Download Logic and Proof Exercises Question 1 Which of the following are true

Logic and Proof Exercises
Question 1
Which of the following are true, and which are false? For the ones which are true, give a
proof. For the ones which are false, give a counterexample.
(a) x is an odd number ⇒ x2 is an odd number.
(b) y is an even number ⇒ y 3 is an even number.
(c) x + y is even ⇒ x and y are both odd.
(d) xy is even ⇒ x and y are both even.
(e) x2 ≥ 0 ⇒ x > 0
(f ) x > 2 ⇒ x2 > 2
(g) x2 ∈ Z ⇒ x ∈ Z
(h) x 6= 0 ∧ y 6= 0 ⇒ xy 6= 0
(i) x 6= 0 ∧ y 6= 0 ⇒ x + y 6= 0
(j) x3 = 8 ⇒ x = 2
Question 2
For each of the statements in question 1, write the converse and state whether it’s true
or not. If true, give a proof. If false, give a counterexample.
Question 3
For each of the statements in question 1, write the contrapositive and state whether it’s
true or not. If true, give a proof. If false, give a counterexample.
Question 4
Use an algebraic proof to prove each of the following true statements.
(a) The product of two odd numbers is an odd number.
(b) The product of two square numbers is a square number. (A square number is an
integer which is the result of squaring another integer. For example, 4 is a square number
because 22 = 4.)
(c) The sum of three consecutive numbers is divisible by three.
(d) The sum of three consecutive even numbers is divisible by six.
(e) The sum of an odd number and an even number is odd.
Question 5
Use proof by contradiction to prove each of the following true statements.
(a)
a b
+ ≥ 2 for all a, b ∈ R
b a
(b) x2 + y 2 ≥ 2xy for all x, y ∈ R
(c) Zero is an even number
(d) There exist no integers a and b for which 18a + 6b = 1.
(e) The square of an odd number is odd.
Question 6
Use the proof that
√
2 is irrational from √
the workshop slides as a guide to construct a
proof that 3 is irrational.
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