Discrete Mathematics Answers

Prove that there is a positive integer that equals the sum of the positive integersnot exceeding it.

Prove that there are infinitely many solutions in positive integers x, y, and z to the equation x^2+y^2=z^2.
Hint: Let x=m^2-n^2 ,y= 2mn, and z = m^2+n^2, where m and n are integers.

in a class of 80 students, 53 study art, 60 study biology, 36 study art and biology, 34 study art and chemistry, 6 study biology only and 18 study biology but bot chemistry. illustrate the information on a venn diagram

(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

(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)