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: 35. For each real number x, let f(x)=x². For any two real numbers a and b, define a -b provided that…

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Q: d. Define a relation R on {1, 2, 3, 4} by R = {(a, b) / a +b is even} and i. Write R as a list of…

A: According to the given information, Let R be a relation on {1,2,3,4} defined as:

Q: Let a and b be integers, at least one of them not 0. Prove that an integer e can be expressed as a…

A: Let a and b be intgers ateast one of them not 0. Our Aim is to prove that an integer e can be…

Q: If 0 ≤ an ≤ b, and Σb, diverges, then Σan diverges.

A: Given: The series an and bn in the interval 0≤an≤bn and if the series ∑bn diverges, then ∑an…

Q: Let RC RX R with {(x, y) |x² =5 y²}. Characterize R in terms of whether it is reflexive,…

A: A reflexive, symmetric, transitive binary relation on a set is an equivalence relation. In other…

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: 2. If A = {1, 2, 3} and B = {2, 3, 4}, and R is the relation on AxB defined by xRy when x + 2y is…

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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: Let ‘E’ & ‘F’ are any two non-empty sets and there order are as follows;…

A: given let ‘E’ & ‘F’ are any two non-empty sets and there order are as follows;…

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: Find a set A and a relation R such that the relation is symmetric, and not reflexive and transitive.

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Q: Let T be the transformation defined by T [1 0 1 -1 2 1 -5 3 -8 A = Ax, where -2 0 -2] a. The domain…

A: Introduction:  the action of the linear transformations was to multiply by a matrix. It turns out…

Q: etermine whether the set of x-2x²,2+5x-x²x+x²} is linerly independent in P

A: First we use definition of linearly independent Set and we check given vectors are linearly…

Q: 1. Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let P = P(A) be the power set of A. Let R be the relation on…

A: (a) For any X ∈ P,            | X ∪ X | = |X|           | X ∪ {1, 2} | = |X| + |{1, 2}| - |X ∩ {1,…

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: Show that if R and R2 are equivalence relations on A, then R1 n R2 is an equivalence relation on A.

A: Equivalence relation on a set A

Q: c) Describe the set of all 7. For a function f:A → B, the statement "f is surjective" is defined by…

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Q: Which of the following is/are true for the relation R, where a is related to b (aRb), if and only if…

A: Suppose, A be any set and suppose R:A->A is a relation. Then, R is reflexive: if for every x in…

Q: Prove the following statements. (a) If f and g are bijective then f-¹ og¯¹ ℃ (gof)¯¹. (b) If f is…

A: This is a question of mapping and set theory.

Q: y) ∈ R mean

A: Given as, A = {4, 5, 6} and B = {5, 6, 7}

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: Determine the orders of the elements of D∞

A: It is known that, D∞=a,b|a2=b2=e also it can be observed that, D∞=abka,abk,bak,bakb Let x∈D∞. If…

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: Let 4= {1, 2, 3, 4, 5, 6} and the relation R on A is defined by R = {(a,b)|a + b is prime; a, b E A}…

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Q: Let A be a set and R and S be relations on A. Dene the relation R + S on A by R + S = (R-S) […

A: 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) ∪…

Q: Prove the following statements. (a) If f and g are bijective then f-1 og ¹ C (gof)-¹. (b) If f is…

A: This is a problem of function.

Q: Show that the set { 159 is linearly independent

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Q: Show that if for a relation R we know that R2 C R, then R transitive, and conversely. Ent. There are…

A: To prove whether a relation is transitive or not

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, B and C be nonempty sets and let f : A → B, g: BC and h: B → C be functions. For each of the…

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Q: Reflexive. • Anti-reflexive. • Symmetric. • Anti-symmetric. Justify all your answers. Transitive. •…

A: Let's consider a few examples of elements inside the relation R, where (a,b)∈ R if and only if…

Q: Let A, B, and C be nonempty sets and let f: A -> B, g: B -> C and h: B -> C be functions. Find a…

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