If ƒ : R → R and g : R → R are both one-to-one, is f + g one-to-one? If f and g are both onto, is f + g onto?

Q: Show that if f : A → B and g : B → C are onto, then gof : A → C is also onto.

A: Given, f : A → B and g : B → C are onto.

Q: Let A={a,b,c,d,e}. Find a relation R on A which is symmetric and transitive but not reflexive

A: Reflexive relation means every term of a set is related with itself

Q: Let S = {[k 1 0],[1 k 1],[0 1 k]}. Deter- mine all the real values of k so that S is a linearly…

A: Given that, S = {[k    1    1], [1     k    1], [0    1    k]} If this set of vectors is linearly…

Q: e) Does the equality |A U B| = |A| + |B| hold for every set A and B? f) Is it true that for any…

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Q: A. Let R be a relation on set A. A={x :xeJ and 0 sx s 12), and R={(a,b) : la - b| is multiple of 4}.…

A: Since not particular question asked, as per guidelines solution to only first question(i.e A) is…

Q: b) Check whether the relation R on the set S = {1, 2, 3} with the the matrix 1 1 1 0 1 1 1 1 1 is an…

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Q: Suppose R and S are symmetric relations on a set A. Prove that RoS is symmetric iff RoS = So R.

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Q: Let A be a set and R and S be relations on A. Define the relation R+S on A by R+S = (R – S)U(S – R).…

A: Given A be a set and R and S be relations on A. R+S on A is defined by R+S=R-S∪S-R=R∪S-R∩S=R∪S∩R∩Sc…

Q: (i) Let A = {1,2,3,4,5), then either find a relation on A that is not reflexive on A and not…

A: Solution:

Q: A. Let R be a relation on set A. A={x : x ∊ J and 0 ≤ x ≤ 12},  and R={(a,b) :  a - b  is multiple…

A: Since not particular question asked ,as per guidelines solution to only first question (i.e A) is…

Q: Suppose f: A → B, f is one-to-one and onto, and Y c B. Show that the inverse image of Y under f and…

A: The function f is one-one and onto.   The function f is one-one if two different element in A…

Q: f is said to be a if it is both one to one and onto. That is, there is a one to one correspondence…

A: f if said to be a------- if it is both one to one and onto.

Q: Suppose f: B → C and g: A → B are functions. Check all the statements that are true: A. If fog is…

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Q: Let X = {1,2,3,4} and Y = {5,6,7}. State whether a function f: X → Y can be one-to-one or not…

A: By using definition of one-time and onto function we solve the given problem :

Q: Let A be the set of all permutations of the 26 letters in the English alphabet. The relation R on A…

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Q: Suppose S and T are infinite sets and φ: S→T is a function that is not onto. May we conclude that S…

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Q: Are f and g both necessarily onto, if gof is onto?

A: Given: f and g are two functions and gof is onto.

Q: y)∈R means that x²-y² is a nonnegative integer. Which of the following is true?     a. 1 R 2 b. 2 R…

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Q: Suppose that each of R ad T is a relation on a set A, the matrix for R is MB, and he matrix for T is…

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Q: With regard to composition of relations, If S and R are both anti-reflexive, then  s∘r is…

A: S&R both are anti- reflexive. Therefore There is no common elements in theses two sets.

Q: Question 11 Given domain of the relation R is {a}. R = {(a, a)} is a reflexive, symmetric and…

A: The domain of relation R is a.

Q: Let X = {1,2,3,4} and Y = {5,6,7}. State whether a function f : X → Y can be one-to-one or not…

A: By using definition of one-time and onto function we solve the given problem :

Q: Let R be a relation on the set of all integers such that aRb if and only if a + b is even. Is R…

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Q: 1. Let A, B, and C be sets such that An Bn C + Ø. Draw a Venn diagram for each of the following set…

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Q: Let A = {x | 5x + 1 E Q} and B be the set of all even integers. Prove that A and B have the same…

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Q: If f:{1,2,3}->{a,b}, f(1)=a f(2)=b f(3)=a then explain why f is onto but not one to one

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Q: 1. Suppose R and S are symmetric relations on a set A. Prove that Ros is symmetric if Ros=SOR

A: Given:  To prove R and S are symmetric relations.

Q: 4. Let the relation R = { (a, b) e Z × Z l(a + b) = 1 (mod 5) }. a. Is the relation transitive?…

A: A relation R between two sets A and B is the set of all ordered pairs a,b such that a∈A and b∈B. We…

Q: Consider the relation on Z × Z given by aRb if a < b. Is R transitive. Answer yes or no…

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Q: . Define a relation R on Z as follows: a Rb if and only if 5 | (a²- 6²). Find four integers which…

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Q: Let R be a relation from A to B and S a relation from B to A. Prove or disprove that if S of R =…

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