Math 220 Assignment 2
Due September 25th
1. Let P , Q, and R be statements. Prove that (P ∧ Q) → R is logically
equivalent to P → (Q → R), using two methods:
(a) using a truth table.
(b) using logical equivalences, such as ¬(P ∧ Q) ≡ ¬P ∨ ¬Q.
2. Let n be an integer. Prove that n is divisible by 6 if and only if n is even
and is divisible by 3.
3. Let n be an integer. Prove that n is disivible by 6 if and only if n2 is
divisible by 6. Hint: you are allowed to use the last problem, and any
other results covered in class.
4. A real number x is rational if there exists integers p and q such that
x = pq . Let x and y be numbers.
(a) Prove or disprove: If x and y are rational, then x + y is rational.
(b) Prove or disprove: If x is rational, and y is irrational, then x + y is
irrational.
(c) (Extra credit): If x and y are both irrational, then x + y is irrational.
You may assume the following true result: If x + y is an integer, then
x and y are integers.
5. Which of the following numbers are rational? Justify your answer. Your
response should be ‘x is rational’, or x is irrational, followed by a proof.
√
(a) 3.
√
(b) 4.
√
(c) 6.
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