1. Let \( A \) and \( B \) be two finite sets. Justify the following statements.(a) If \( f: A \rightarrow B \) is an injection, then \( |A| \leq |B| \). (Consider the Pigeonhole Principle.)(b) If \( f: A \rightarrow B \) is a surjection, then there is an injection \( g: B \rightarrow A \), and deduce from the first part that \( |B| \leq |A| \).

To prove if f: A→B is injective then A≤B It is sufficient to prove that if A>B then f is not…

Answered: 1. Let A and B be two finite sets.…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Answer this question for me mate. Appreciate it. :)
Exercise 3.3.15. Prove or find a counterexample to the following statement. Let A, B and C be sets. Then (A-B) UC = (AUBUC) - (ANB).
Let Vi = x ℝ − 1 i ≤ x ≤ 1 i = − 1 i , 1 i for each positive integer i. Find each of the following. (Enter your answers using interval notation.) (a) ∪4i = 1Vi = (b) ∩4i = 1Vi = (c) Are V1, V2, V3, mutually disjoint? Explain. a.) Yes, because the intersection of the sets V1, V2, V3, ... is empty. b.) Yes, because no two of the sets V1, V2, V3, ... have any elements in common. c.) Yes, because the union of the sets V1, V2, V3, ... is empty. d.) No, because no two of the sets V1, V2, V3, ... are disjoint. e.) No, because the sets V1, V2, V3, ... are disjoint. (d) ∪ni = 1Vi = (e) ∩ni = 1Vi = (f) ∪∞i = 1Vi = (g) ∩∞i = 1Vi =
Solve number 2
c) Let the set A = {-1,0, 1} and consider the following function: f3 A × A → A × A given by f(x,y) = (y7,x). Decide whether this function is injective, surjective, a bijection. f3 is an injection: Not Answered • ƒ3 is a surjection: Not Answered f3 is a bijection: Not Answered d) Let the set A = {1, 2, 3,4} and let P(A) denote the power-set of A. Consider the following function: f₁ : P(A) → P(A) given by ƒ(B) = {1, 2, 3}\B. Decide whether this function is injective, surjective, a bijection. f4 is an injection: Not Answered f4 is a surjection: Not Answered • f is a bijection: Not Answered
Q. Let flx)- if x20 07 xt Prove that fX) = 2|X| for r every X X

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1. Let A and B be two finite sets. Justify the following statements. (a) If f: A B is an injection, then |A|≤|B|. (Consider the Pigeonhole Principle.) (b) If f: AB is a surjection, then there is an injection g: BA, and deduce from the first part that B|≤|A|.