Discrete Mathematics Answers

(1) Let Q(x, y) be the statement "x+y=x-y." If the domain for both variables consists of all integers, what are the truth values?
(a)Q(1,1)
(b)Q(2,0)
(c)∀yQ(1, y)
(d)∃xQ(x,2)
(e)∃x∃yQ(x, y)
(f)∀x∃yQ(x, y)
(g)∃y∀xQ(x, y)
(h)∀y∃xQ(x, y)
(i)∀x∀yQ(x, y)

Give an example to prove that intersection of denumerable sets need not be denumerable

Prove that 3Z is an infinite set

(8) Determine whether the argument below is valid:
"If Superman were able and willing to prevent evil, he would do so. If Superman were
unable to prevent evil, he would be impotent; if he were unwilling to prevent evil, he
would be malevolent. Superman does not prevent evil. If Superman exists, he is neither
impotent nor malevolent. Therefore Superman does not exist."

(7) The Logic Problem has two assumptions:
(i) "Logic is difficult or not many students like logic."
(ii) "If mathematics is easy, then logic is not difficult."
By translating these assumptions into statements involving propositional variables and
logical connectives, determine whether each of the following are valid conclusions of these
assumptions:
(a) That mathematics is not easy, if many students like logic.
(b) That not many students like logic, if mathematics is not easy.
(c) That mathematics is not easy or logic is difficult.
(d) That logic is not difficult or mathematics is not easy.
(e) That if not many students like logic, then either mathematics is not easy or logic is
not difficult.

(6) Determine the truth value of the statement ∃x∀y(x lessthanorequalto y2) if the domain for the
variables consists of
(a) The positive real numbers.
(b) The integers.
(c) The nonzero real numbers.

(5) Find a counterexample, if possible, to these universally quantified statements, where
the domain for all variables consists of all integers.
(a)∀x∃y(x= 1/y)
(b)∀x∃y(y^2-x <100)
(c)∀x∀y(x^2=/=y^3)

(4) Find a common domain for the variables x, y, and z for which the statement
∀x∀y((x=/=y) => ∀z((z=x)v(z=y)))
is true and another domain for which it is false.

Determine the truth value of each of ∀x∀y∃z(z=(x+y)/2)

1. Let U = {l, 2, 3, 4, 5, 6, 7, 8, 9, and 10} be a universal set. Let A, B, C such that A= {l, 3, 4, 8},
B = {2, 3, 4, 5, 9, 10}, and C = {3, 5, 7, 9, 10}. Use bit representations
for A, B, and C together with UNION,
INTER, DIFF, and COMP to find the bit representation for the following:
(a) AU B
(b) An B n C
(f)) (AU C) n B
(d) (A - B) UC
(e) An (B - (C n B))
(f) A - (B - C)
(g) (AU B) U (C - B)