Question

Perform the following tasks:

a. Prove directly that the product of an even and an odd number is even.

b. Prove by contraposition for arbitrary x does not equal -2: if x is irrational, then so is x/(x+2)

c. Disprove: If x is irrational and y is irrational, then x+y is irrational.

Answer #1

Prove or disprove the following statements. Remember to disprove
a statement you have to show that the statement is false.
Equivalently, you can prove that the negation of the statement is
true. Clearly state it, if a statement is True or False. In your
proof, you can use ”obvious facts” and simple theorems that we have
proved previously in lecture.
(a) For all real numbers x and y, “if x and y are irrational,
then x+y is irrational”.
(b) For...

(a) Prove or disprove the statement (where n is an integer): If
3n + 2 is even, then n is even.
(b) Prove or disprove the statement: For irrational numbers x
and y, the product xy is irrational.

Consider the following statement:
If x and y are integers and x - y is odd, then x is odd or y is
odd.
Answer the following questions about this statement.
2(a) Provide the predicate for the starting assumption for a
proof by contraposition for the given statement.
2(b) Provide the conclusion predicate for a proof by
contraposition for the given statement.
2(c) Prove the statement is true by contraposition.
2(d) Prove that the converse is not true.

(b) If n is an arbitrary element of Z, prove directly that n is
even iff n + 1 is odd. iff is read as “if and only if”

Write the contrapositive statements to each of the following.
Then prove each of them by proving their
respective contrapositives. In both statements assume x
and y are integers.
a. If the product xy is even, then at least one
of the two must be even.
b. If the product xy is odd, then both x and y
must be odd.
3. Write the converse the following statement.
Then prove or disprove that converse depending on whether it is
true or not. Assume x...

Theorem: If m is an even number and n is an odd number, then
m^2+n^2+1 is even. Don’t prove it.
In writing a proof by contraposition, what is your “Given”
(assumption)? ___________________________
What is “To Prove”: _____________________________

3.a) Let n be an integer. Prove that if n is odd, then
(n^2) is also odd.
3.b) Let x and y be integers. Prove that if x is even and y is
divisible by 3, then the product xy is divisible by 6.
3.c) Let a and b be real numbers. Prove that if 0 < b < a,
then (a^2) − ab > 0.

Let x, y ∈Z. Prove that (x+1)y^2 is even if and only if x is odd
and y is even.

Prove the following statements by contradiction
a) If x∈Z is divisible by both even and odd integer, then x is
even.
b) If A and B are disjoint sets, then A∪B = AΔB.
c) Let R be a relation on a set A. If R = R−1, then R is
symmetric.

(1) Let x be a rational number and y be an irrational. Prove
that 2(y-x) is irrational
a) Briefly explain which proof method may be most appropriate to
prove this statement. For example either contradiction,
contraposition or direct proof
b) State how to start the proof and then complete the proof

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