Other questions and answers in April 2020 — Page 113

Let f :A→B. Prove that f ◦ IA = IB ◦ f = f, where IA, IB are the identity functions on A, B, respectively. Prove the following theorem: Let f : X → Y . Then f is invertibl...

Prove that for sets A and B, A^C∩B^C=(A∪B)^C.

Let nZ represent the set of integers divisible by n. Prove that (ab)Z⊆aZ∩bZ. Give a counterexample to the statement aZ∩bZ⊆(ab)Z.

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Find the gcd of 309876 and 191352. Then find all the integers x and y such that 309876x + 191352y = 19992. Please write the procedures.

Calculate 2^5937 (mod 40). Prove that if nx == nb (mod nm), then x == b (mod m), and use this to find all the solutions to 6x + 3 == 21 (mod 24). Your answer should be a list of numbers (mod 24).

What brought about the Crusades initially and why were continued in the Third and Fourth Crusades? (Be sure to consider the roles of emperors of the Byzantine Empire, Pope Urban II, and later religiou...

Let A = {{1,2}, {1,3}, {2}, {2,5}, {3,4,5}}. Define f: A→N by f(a)= the sum of the elements of a (∀a∈A) Find f[A]. Now define g : N → N by the largest prime ≤n if n > 1 g(x) ...

Assume that f: X→Y and g: Y →Z are both functions. For each of the following statements, if the statement is true, then prove it. If the statement is false, provide a counterexample. 1. If g ?...

Let f: A→B, g:B→C, h:C→D. Prove that h ◦ (g ◦ f ) = (h ◦ g) ◦ f. (Associative law) Combine the two theorems to write the corollary. Theorem #1. If f : X → Y and g :?...

Consider the relation S on a set of real numbers R as: Is S Reflexive? Why? Is S Symmetric? Why? Is S Transitive? Why?

1.Define f: R→R by f(x)= sinx and g: R→R by g(x)= 2x 2 + 1. Find f◦g and g◦f, and identify the range in each case. 2.Define f: R→R and g: R→R by f(x) = x 2 x+1, if x > 0 g(x)={ -10, i...

Use mathematical induction to prove the statement: