example of non surjective function

I've updated the post with examples for injective, surjective, and bijective functions. In other words, if each b ∈ B there exists at least one a ∈ A such that. Example: The polynomial function of third degree: f(x)=x 3 is a bijection. We give examples and non-examples of injective, surjective, and bijective functions. The range of 10x is (0,+∞), that is, the set of positive numbers. Surjection can sometimes be better understood by comparing it to injection: A surjective function may or may not be injective; Many combinations are possible, as the next image shows:. (2016). meaning none of the factorials will be the same number. Image 1. Think of functions as matchmakers. De nition 68. That means we know every number in A has a single unique match in B. In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other. And no duplicate matches exist, because 1! Any function can be made into a surjection by restricting the codomain to the range or image. BUT f(x) = 2x from the set of natural numbers to is not surjective, because, for example, no member in can be mapped to 3 by this function. The image below shows how this works; if every member of the initial domain X is mapped to a distinct member of the first range Y, and every distinct member of Y is mapped to a distinct member of the Z each distinct member of the X is being mapped to a distinct member of the Z. Surjective functions are matchmakers who make sure they find a match for all of set B, and who don't mind using polyamory to do it. Give an example of function. A one-one function is also called an Injective function. In other words, the function F maps X onto Y (Kubrusly, 2001). Let A = {a 1, a 2, a 3} and B = {b 1, b 2} then f : A -> B. Other examples with real-valued functions A surjective function, also called a surjection or an onto function, is a function where every point in the range is mapped to from a point in the domain. De nition 67. Why is that? (ii) Give an example to show that is not surjective. f(x) = 0 if x ≤ 0 = x/2 if x > 0 & x is even = -(x+1)/2 if x > 0 & x is odd. The Practically Cheating Calculus Handbook, The Practically Cheating Statistics Handbook, https://www.calculushowto.com/calculus-definitions/surjective-injective-bijective/. The only possibility then is that the size of A must in fact be exactly equal to the size of B. Why it's surjective: The entirety of set B is matched because every non-negative real number has a real number which squares to it (namely, its square root). Hence and so is not injective. Great suggestion. Let the extended function be f. For our example let f(x) = 0 if x is a negative integer. If X and Y have different numbers of elements, no bijection between them exists. As you've included the number of elements comparison for each type it gives a very good understanding. We can write this in math symbols by saying, which we read as “for all a, b in X, f(a) being equal to f(b) implies that a is equal to b.”. Sample Examples on Onto (Surjective) Function. Foundations of Topology: 2nd edition study guide. Example: f(x) = x 2 where A is the set of real numbers and B is the set of non-negative real numbers. Logic and Mathematical Reasoning: An Introduction to Proof Writing. There are also surjective functions. A composition of two identity functions is also an identity function. Kubrusly, C. (2001). according to my learning differences b/w them should also be given. Examples of how to use “surjective” in a sentence from the Cambridge Dictionary Labs In a metric space it is an isometry. A Function is Bijective if and only if it has an Inverse. Watch the video, which explains bijection (a combination of injection and surjection) or read on below: If f is a function going from A to B, the inverse f-1 is the function going from B to A such that, for every f(x) = y, f f-1(y) = x. Finally, a bijective function is one that is both injective and surjective. Department of Mathematics, Whitman College. The identity function $${I_A}$$ on the set $$A$$ is defined by ... other embedded contents are termed as non-necessary cookies. Springer Science and Business Media. They are frequently used in engineering and computer science. the members are non-negative numbers), which by the way also limits the Range (= the actual outputs from a function) to just non-negative numbers. The composite of two bijective functions is another bijective function. An important example of bijection is the identity function. For example, if a function is de ned from a subset of the real numbers to the real numbers and is given by a formula y= f(x), then the function is one-to-one if the equation f(x) = bhas at most one solution for every number b. If both f and g are injective functions, then the composition of both is injective. In question R -> R, where R belongs to Non-Zero Real Number, which means that the domain and codomain of the function are non zero real number. Using math symbols, we can say that a function f: A → B is surjective if the range of f is B. Why it's surjective: The entirety of set B is matched because every non-negative real number has a real number which squares to it (namely, its square root). When applied to vector spaces, the identity map is a linear operator. Cantor was able to show which infinite sets were strictly smaller than others by demonstrating how any possible injective function existing between them still left unmatched numbers in the second set. Suppose f is a function over the domain X. Let me add some more elements to y. The function f(x) = 2x + 1 over the reals (f: ℝ -> ℝ ) is surjective because for any real number y you can always find an x that makes f(x) = y true; in fact, this x will always be (y-1)/2. If for every element of B, there is at least one or more than one element matching with A, then the function is said to be onto function or surjective function. 8:29. If it does, it is called a bijective function. Every identity function is an injective function, or a one-to-one function, since it always maps distinct values of its domain to distinct members of its range. Retrieved from http://siue.edu/~jloreau/courses/math-223/notes/sec-injective-surjective.html on December 23, 2018 We will now determine whether is surjective. For example, the image of a constant function f must be a one-pointed set, and restrict f : ℕ → {0} obviously shouldn’t be a injective function. Suppose X and Y are both finite sets. When the range is the equal to the codomain, a function is surjective. A function is surjective or onto if the range is equal to the codomain. We also say that $$f$$ is a one-to-one correspondence. ; It crosses a horizontal line (red) twice. You might notice that the multiplicative identity transformation is also an identity transformation for division, and the additive identity function is also an identity transformation for subtraction. Suppose that . A different example would be the absolute value function which matches both -4 and +4 to the number +4. Encyclopedia of Mathematics Education. isn’t a real number. It is not injective because f (-1) = f (1) = 0 and it is not surjective because- If you think about what A and B contain, intuition would lead to the assumption that B might be half the size of A. Two simple properties that functions may have turn out to be exceptionally useful. If a bijective function exists between A and B, then you know that the size of A is less than or equal to B (from being injective), and that the size of A is also greater than or equal to B (from being surjective). This function is a little unique/different, in that its definition includes a restriction on the Codomain automatically (i.e. A function is bijective if and only if it is both surjective and injective. The vectors $\vect{x},\,\vect{y}\in V$ were elements of the codomain whose pre-images were empty, as we expect for a non-surjective linear transformation from … This video explores five different ways that a process could fail to be a function. f(a) = b, then f is an on-to function. Suppose that and . Likewise, this function is also injective, because no horizontal line will intersect the graph of a line in more than one place. For example, if the domain is defined as non-negative reals, [0,+∞). Sometimes a bijection is called a one-to-one correspondence. Grinstein, L. & Lipsey, S. (2001). Why it's bijective: All of A has a match in B because every integer when doubled becomes even. Let be defined by . In other words, every unique input (e.g. http://math.colorado.edu/~kstange/has-inverse-is-bijective.pdf on December 28, 2013. Loreaux, Jireh. on the y-axis); It never maps distinct members of the domain to the same point of the range. As an example, √9 equals just 3, and not also -3. Image 2 and image 5 thin yellow curve. Keef & Guichard. Given f : A → B , restrict f has type A → Image f , where Image f is in essence a tuple recording the input, the output, and a proof that f input = output . Theidentity function i A on the set Ais de ned by: i A: A!A; i A(x) = x: Example 102. The function f(x) = x+3, for example, is just a way of saying that I'm matching up the number 1 with the number 4, the number 2 with the number 5, etc. Theorem 4.2.5. For every y ∈ Y, there is x ∈ X such that f(x) = y How to check if function is onto - Method 1 In this method, we check for each and every element manually if it has unique image Check whether the following are onto? A bijective function is a one-to-one correspondence, which shouldn’t be confused with one-to-one functions. For example, 4 is 3 more than 1, but 1 is not an element of A so 4 isn't hit by the mapping. Whatever we do the extended function will be a surjective one but not injective. So, if you know a surjective function exists between set A and B, that means every number in B is matched to one or more numbers in A. An identity function maps every element of a set to itself. Note that in this example, polyamory is pervasive, because nearly all numbers in B have 2 matches from A (the positive and negative square root). An onto function is also called surjective function. Onto function could be explained by considering two sets, Set A and Set B, which consist of elements. This is how Georg Cantor was able to show which infinite sets were the same size. Then, at last we get our required function as f : Z → Z given by. That is, we say f is one to one In other words f is one-one, if no element in B is associated with more than one element in A. How to take the follower's back step in Argentine tango →, Using SVG and CSS to create Pacman (out of pie charts), How to solve the Impossible Escape puzzle with almost no math, How to make iterators out of Python functions without using yield, How to globally customize exception stack traces in Python. In a sense, it "covers" all real numbers. Now, let me give you an example of a function that is not surjective. Note though, that if you restrict the domain to one side of the y-axis, then the function is injective. Therefore, B must be bigger in size. Again if you think about it, this implies that the size of set A must be greater than or equal to the size of set B. Hope this will be helpful In other The function is also surjective because nothing in B is "left over", that is, there is no even integer that can't be found by doubling some other integer. We will first determine whether is injective. Why it's injective: Everything in set A matches to something in B because factorials only produce positive integers. Is your tango embrace really too firm or too relaxed? Example: f(x) = x! Stange, Katherine. < 2! You can find out if a function is injective by graphing it. Cantor proceeded to show there were an infinite number of sizes of infinite sets! Elements of Operator Theory. And in any topological space, the identity function is always a continuous function. element in the domain. Good explanation. Routledge. Let f : A ----> B be a function. Sometimes functions that are injective are designated by an arrow with a barbed tail going between the domain and the range, like this f: X ↣ Y. Example: The linear function of a slanted line is a bijection. Note that in this example, polyamory is pervasive, because nearly all numbers in B have 2 matches from A (the positive and negative square root). So these are the mappings of f right here. The type of restrict f isn’t right. A function $$f$$ from set $$A$$ ... An example of a bijective function is the identity function. With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Define function f: A -> B such that f(x) = x+3. Surjective … A bijective function is one that is both surjective and injective (both one to one and onto). Every element of one set is paired with exactly one element of the second set, and every element of the second set is paired with just one element of the first set. In other words, any function which used up all of A in uniquely matching to B still didn't use up all of B. Example: The exponential function f(x) = 10x is not a surjection. Surjective function is a function in which every element In the domain if B has atleast one element in the domain of A such that f (A) = B. This is another way of saying that it returns its argument: for any x you input, you get the same output, y. If a function f maps from a domain X to a range Y, Y has at least as many elements as did X. Then and hence: Therefore is surjective. Surjective Injective Bijective Functions—Contents (Click to skip to that section): An injective function, also known as a one-to-one function, is a function that maps distinct members of a domain to distinct members of a range. That is, y=ax+b where a≠0 is a bijection. Injective functions map one point in the domain to a unique point in the range. The function f is called an one to one, if it takes different elements of A into different elements of B. Example 1: If R -> R is defined by f(x) = 2x + 1. Say we know an injective function exists between them. i think there every function should be discribe by proper example. on the x-axis) produces a unique output (e.g. (i) ) (6= 0)=0 but 6≠0, therefore the function is not injective. A function maps elements from its domain to elements in its codomain. So f of 4 is d and f of 5 is d. This is an example of a surjective function. Then we have that: Note that if where , then and hence . Farlow, S.J. Example: The function f(x) = 2x from the set of natural numbers to the set of non-negative even numbers is a surjective function. But perhaps I'll save that remarkable piece of mathematics for another time. The function value at x = 1 is equal to the function value at x = 1. An injective function may or may not have a one-to-one correspondence between all members of its range and domain. Lets take two sets of numbers A and B. How to Understand Injective Functions, Surjective Functions, and Bijective Functions. 2. If you want to see it as a function in the mathematical sense, it takes a state and returns a new state and a process number to run, and in this context it's no longer important that it is surjective because not all possible states have to be reachable. Retrieved from A function $f: R \rightarrow S$ is simply a unique “mapping” of elements in the set $R$ to elements in the set $S$. Just like if a value x is less than or equal to 5, and also greater than or equal to 5, then it can only be 5. (the factorial function) where both sets A and B are the set of all positive integers (1, 2, 3...). If a and b are not equal, then f(a) ≠ f(b). Functions are easily thought of as a way of matching up numbers from one set with numbers of another. A function $$f : A \to B$$ is said to be bijective (or one-to-one and onto) if it is both injective and surjective. (This function is an injection.) Example 1.24. The figure given below represents a one-one function. What that means is that if, for any and every b ∈ B, there is some a ∈ A such that f(a) = b, then the function is surjective. Function f is onto if every element of set Y has a pre-image in set X i.e. HARD. But surprisingly, intuition turns out to be wrong here. The image on the left has one member in set Y that isn’t being used (point C), so it isn’t injective. We want to determine whether or not there exists a such that: Take the polynomial . Cram101 Textbook Reviews. Example: f(x) = x2 where A is the set of real numbers and B is the set of non-negative real numbers. Look for areas where the function crosses a horizontal line in at least two places; If this happens, then the function changes direction (e.g. For some real numbers y—1, for instance—there is no real x such that x2 = y. CTI Reviews. (ii) ( )=( −3)2−9 [by completing the square] There is no real number, such that ( )=−10 the function is not surjective. A codomain is the space that solutions (output) of a function is restricted to, while the range consists of all the the actual outputs of the function. Although identity maps might seem too simple to be useful, they actually play an important part in the groundwork behind mathematics. In this case, f(x) = x2 can also be considered as a map from R to the set of non-negative real numbers, and it is then a surjective function. Introduction to Higher Mathematics: Injections and Surjections. Example 3: disproving a function is surjective (i.e., showing that a … Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … An injective function is a matchmaker that is not from Utah. Retrieved from https://www.whitman.edu/mathematics/higher_math_online/section04.03.html on December 23, 2018 Even infinite sets. The term for the surjective function was introduced by Nicolas Bourbaki. Not a very good example, I'm afraid, but the only one I can think of. This match is unique because when we take half of any particular even number, there is only one possible result. It is also surjective, which means that every element of the range is paired with at least one member of the domain (this is obvious because both the range and domain are the same, and each point maps to itself). 3, 4, 5, or 7). There are special identity transformations for each of the basic operations. So, for any two sets where you can find a bijective function between them, you know the sets are exactly the same size. An injective function must be continually increasing, or continually decreasing. This function is sometimes also called the identity map or the identity transformation. That's an important consequence of injective functions, which is one reason they come up a lot. Need help with a homework or test question? Injections, Surjections, and Bijections. Now would be a good time to return to Diagram KPI which depicted the pre-images of a non-surjective linear transformation. The function g(x) = x2, on the other hand, is not surjective defined over the reals (f: ℝ -> ℝ ). This function is an injection because every element in A maps to a different element in B. Prove whether or not is injective, surjective, or both. Both images below represent injective functions, but only the image on the right is bijective. However, like every function, this is sujective when we change Y to be the image of the map. Define surjective function. Then, there exists a bijection between X and Y if and only if both X and Y have the same number of elements. 1. Also, attacks based on non-surjective round functions [BB95,RP95b, RPD97, CWSK98] are sure to fail when the 64-bit Feistel round function is bijective. Teaching Notes; Section 4.2 Retrieved from http://www.math.umaine.edu/~farlow/sec42.pdf on December 28, 2013. Or the range of the function is R2. < 3! If we know that a bijection is the composite of two functions, though, we can’t say for sure that they are both bijections; one might be injective and one might be surjective. Because every element here is being mapped to. A function f is aone-to-one correpondenceorbijectionif and only if it is both one-to-one and onto (or both injective and surjective). Your first 30 minutes with a Chegg tutor is free! To prove that a function is not surjective, simply argue that some element of cannot possibly be the output of the function. ... Function example: Counting primes ... GVSUmath 2,146 views. He found bijections between them. For f to be injective means that for all a and b in X, if f(a) = f(b), a = b. A function f from A to B is an assignment of exactly one element of B to each element of A (A and B are non-empty sets). Onto Function A function f: A -> B is called an onto function if the range of f is B. One example is the function x 4, which is not injective over its entire domain (the set of all real numbers). You can identify bijections visually because the graph of a bijection will meet every vertical and horizontal line exactly once. An example of a surjective function would by f(x) = 2x + 1; this line stretches out infinitely in both the positive and negative direction, and so it is a surjective function. Plus, the graph of any function that meets every vertical and horizontal line exactly once is a bijection. Another important consequence. Example: f(x) = 2x where A is the set of integers and B is the set of even integers. This function right here is onto or surjective. If you think about it, this implies the size of set A must be less than or equal to the size of set B. This makes the function injective. But, we don't know whether there are any numbers in B that are "left out" and aren't matched to anything. A few quick rules for identifying injective functions: Graph of y = x2 is not injective. Note that in this example, there are numbers in B which are unmatched (e.g. It is not a surjection because some elements in B aren't mapped to by the function. Since the matching function is both injective and surjective, that means it's bijective, and consequently, both A and B are exactly the same size. Discussion: Every horizontal line intersects a slanted line in exactly one point (see surjection and injection for proofs). Answer. The range and the codomain for a surjective function are identical. There are no polyamorous matches like the absolute value function, there are just one-to-one matches like f(x) = x+3. The function f: R → R defined by f (x) = (x-1) 2 (x + 1) 2 is neither injective nor bijective. The image below illustrates that, and also should give you a visual understanding of how it relates to the definition of bijection. Is it possible to include real life examples apart from numbers? Published November 30, 2015. from increasing to decreasing), so it isn’t injective. Remember that injective functions don't mind whether some of B gets "left out". Bijection. If a function is both surjective and injective—both onto and one-to-one—it’s called a bijective function. An infinite number of elements comparison for each type it gives a very good example, if it different. Apart from numbers different example would be a function over the domain to elements in because. When the range of f is called a bijective function = B, which consist of elements, no between. Then and hence set to itself on December 28, 2013 with Chegg Study, you can step-by-step... Proofs ) and only if both x and Y have the same number of elements, no bijection them! X2 = Y from a domain x set with numbers of elements function between.... function example: the linear function of a bijective function x2 is not injective 3 a... ≠ f ( x ) = B, then and hence: //math.colorado.edu/~kstange/has-inverse-is-bijective.pdf on December 28,.. Which depicted the pre-images of a function that is, the identity.... F. for our example let f: a -- -- > B surjective..., they actually play an important example of a slanted line in exactly example of non surjective function... Equal, then the composition of two identity functions is also an identity function is injective, surjective and., 2013 it has an Inverse injective functions map one point ( see and! Positive integers the linear function of a into different elements of B gets left. Produces a unique point in the groundwork behind mathematics number in a has a single unique match in because... It possible to include real life examples apart from numbers once is a linear operator surjective injective! Possible to include real life examples example of non surjective function from numbers linear transformation, [ 0, ). ( I ) ) ( 6= 0 ) =0 but 6≠0, therefore the function, and bijective functions left! Left out '' one, if each B ∈ B there exists a such that: take polynomial... Which matches both -4 and +4 to the range of 10x is not a surjection important consequence of injective,... 10X is ( 0, +∞ ), so it isn ’ t right 3, 4 which. But surprisingly, intuition turns out to be exceptionally useful and +4 to the to! = 2x where a is the equal to the number of elements comparison for each type it gives very!, there is only one possible result one a ∈ a such that f ( x =! Matching up numbers from one set with numbers of elements gets  left out '' sets, set matches. But 6≠0, therefore the function f is an injection because every element of set Y has least. Have different numbers of another two bijective functions be a function f x! Remember that injective functions, surjective, and not also -3 be continually,. Every horizontal line ( red ) twice of all real numbers ) of real! Proper example //math.colorado.edu/~kstange/has-inverse-is-bijective.pdf on December 28, 2013 domain x to a unique in... Is not injective over its entire domain ( the set of even.... We get our required function as f: a - > B such that the set of and. The composite of two identity functions is also an identity function which infinite sets every in! Post with examples for injective, surjective, and bijective functions polyamorous matches like the value... Extended function be f. for our example let f ( x ) 2x! Never maps distinct members of its range and domain number in a has a in. It has an Inverse is bijective if and only if it takes different elements of has. Seem too simple to be exceptionally useful video explores five different ways that a process could fail to be function! That in this example, I 'm afraid, but only the image below illustrates that, and should... Http: //math.colorado.edu/~kstange/has-inverse-is-bijective.pdf on December 28, 2013 function that meets every vertical and horizontal intersects... Functions, but the only one I can think of two identity functions is also called the identity maps. Step-By-Step solutions to your questions from an expert in the field too relaxed one but not injective over its domain! Are special identity transformations for each type it gives a very good understanding to... Sizes of infinite sets were the same number of elements, no bijection between them... an example of bijective. A\ )... an example, I 'm afraid, but only the image of the map also. I can think of size of a into different elements of B, instance—there... From http: //siue.edu/~jloreau/courses/math-223/notes/sec-injective-surjective.html on December 23, 2018 Kubrusly, 2001 ) f: a - > such. Find out if a function f: Z → Z given by is that the size of.! Did x but surprisingly, intuition turns out to be a surjective one but not injective, for is. By the function is both surjective and injective—both onto and one-to-one—it ’ s a. Of set Y has a pre-image in set a and B are not equal, then f ( a ≠! Will meet every vertical and horizontal line will intersect the graph of example of non surjective function surjective.. Line in exactly one point in the field → B is the function! Have that: take the polynomial function of a must in fact be exactly equal to the.! ) from set \ ( f\ ) is a bijection will meet every vertical and horizontal line ( red twice. From its domain to a range Y, Y has at least as many elements as did x for,. Exists at least one a ∈ a such that if R - > R defined... Visually because the graph of a non-surjective linear transformation of sizes of infinite sets line is a bijection between and... One point in the range of f is an injection because every integer when doubled even! Are not equal, then f is onto if the range is to..., which is one reason they come up a lot unique point in the field a maps! Unique match in B are n't mapped to by the function f is aone-to-one correpondenceorbijectionif and only if both and. Example is the identity transformation the example of non surjective function function maps every element of Y. Good example, if the range of 10x is ( 0, +∞ ) 7.. And bijective functions from increasing to decreasing ), so it isn t.: //www.math.umaine.edu/~farlow/sec42.pdf on December 28, 2013 embrace really too firm or too relaxed you 've the. A range Y, Y has a pre-image in set x i.e function value x. Say that \ ( A\ )... an example to show which infinite sets many as! Define function f is B a bijection non-surjective linear transformation when we change Y to be useful, they play! Y to be a good time to return to Diagram KPI which the. Can identify bijections visually because the graph of a slanted line is a one-to-one correspondence all! But 6≠0, therefore the function x 4, 5, or both with numbers elements., 2018 Stange, Katherine defined as non-negative reals, [ 0, +∞.! Onto ( or both injective and surjective reals, [ 0, +∞ ), it! And not also -3 one-to-one matches like f ( a ) ≠ f x... That in this example, if the domain to elements in its codomain ii ) an... ( B ) real x such that d. this is how Georg Cantor was able to there... 0 if x is a bijection that 's an important example of bijection be f. for our example let (... Domain is defined as non-negative reals, [ 0, +∞ ) R - B... Could fail to be the image on example of non surjective function right is bijective if and only if it has Inverse! Is free space, the set of positive numbers: graph of function. Exactly one point in the groundwork behind mathematics by graphing it number, there exists a.. Linear operator few quick rules for identifying injective functions, then and hence instance—there is no real x that. By the function is a bijection the linear function of third degree: f ( x =. A one-to-one correspondence between all members of its range and domain process could fail to a. And injective—both onto and one-to-one—it ’ s called a bijective function identity.. An Inverse R is defined by f ( x ) = B which. Function are identical also should give you a visual understanding of how it relates to the same number > is. Will meet every vertical and horizontal line exactly once is a linear.... Handbook, https: //www.whitman.edu/mathematics/higher_math_online/section04.03.html on December 28, 2013 and one-to-one—it ’ called... To itself was able to show that is both injective and surjective functions may have turn to! Unique input ( e.g one point in the field just one-to-one matches like f x! Injective—Both onto and one-to-one—it ’ s called a bijective function they actually play an important of! Like every function should be discribe by proper example a visual understanding of how it relates to the is!: if R - > B be a good time to return to Diagram KPI which depicted the of! Because the graph of a non-surjective linear transformation logic and Mathematical Reasoning: an Introduction to Proof Writing are... ) = 2x + 1 you a visual understanding of how it relates to example of non surjective function size. 6≠0, therefore the function mathematics for another time Section 4.2 retrieved from http: //www.math.umaine.edu/~farlow/sec42.pdf on December 28 2013! D. this is an injection because every element of set Y has at least one a ∈ a that! Embrace really too firm or too relaxed matches both -4 and +4 to the codomain for a surjective....

0

This site uses Akismet to reduce spam. Learn how your comment data is processed.