Discrete Math Problems

Susanne Epp, Discrete Math An introduction to mathematical reasoning, brief edition..
section 6.3

find a counterexample..
(2) for all sets A and B, (A UNION B)ᶜ = Aᶜ UNION Bᶜ

prove if it is true, or find a counterexample if it is false
(7) For all sets A, B, C, (A-B) INTERSECT (C-B) = A - (B UNION C).

construct an algebraic proof for the given statement. cite a property from theorem 6.2.2 for every step
(34) For all sets A, B, C, (A-B) - C = A- (B UNION C)

(37) For all sets A and B, (Bᶜ UNION (bᶜ - A ))ᶜ = B

Simplify the given expression, cite a property from theorem 6.2.2 for each step.

(41) A INTERSECT ((B UNION Aᶜ) INTERSECT Bᶜ)

Section 1.3
(12) Define a relation T from R to R as follows: For all real numbers x and y, (x, y) in T means that y² - x² = 1. Is T a function? Explain.


(20)

Define functions H and K from R to R by the following formulas: For all x in R, H(x) = (x-2)² and K(x) = (x-1)(x-3)+1. Does H=K? Explain.

Section 7.1

(16) Let F and G be functions from the set of all real numbers to itself. Define new functions F-G: R->R and G-F: R->R as follows: For all x in R,

(F-G)(x) = F(x) - G(x)
(G-F)(x) = G(x) - F(x)

Does F-G = G-F? Explain.

(28)
Student C tries to define a function h: Q-> Q by the rule h(m/n) = (m²)/n for all integers m and n with n != 0. Student D claims that h is not well defined. Justify student D's claim.

(32)
Let X= {1,2,3,4} and Y= {a,b,c,d,e}. Define g: X->Y as follows: g(1) = a, g(2) = a, g(3) = a, and g(4) = d.

(a) draw an arrow diagram for g.
(b) Let A = {2,3}, C={a}, and D={b,c}. Find g(A), g(X), g⁻¹(C), g⁻¹(D), and g⁻¹(Y)

let X and Y be sets, let A and B be any subsets of X, and let C and D be any subsets of Y. Determine which of the properties are true for all function F from X to Y and which are false for at least one function F from X to Y. Justify answers.

(37)
For all subsets A and B of X, F(A-B) = F(A) - F(B).

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