1. Apply a truth table to show each conclusion of following: (a) ~(~p) ≡ p (b) ~(p ∨ ?) ≡ (~?) ∧ (~?)
1. Apply a truth table to show each conclusion of following:
(a) ~(~p) ≡ p (b) ~(p ∨ ?) ≡ (~?) ∧ (~?)
2. Write down the converse, inverse and contra-positive of each of the following statements:
(a) For any real number
x
, if
x 4,
then
2
x 16 .
(b) If both a and b are integers, then their product ab is an integer.
3. Use logical equivalences to simplify each one of following
a) ((? ∧ ¬?) ∨ (? ∧ ?)) ∧ ? (b)
((p q)(p q))(p q)
4. Negating the following statements:
(a)
primes p, p is odd.
(b)
a triangle T such that the sum of the angles equals
0
200 .
(c) For every square x there is a triangle y such that x and y have different colors.
(d) There exists a triangle y such that for every square x, x and y have different colors.
(e)
people p, if p is blond then p has blue eyes.
5. Construct a truth table to determine whether or not the argument is valid
(a) ? ∨ (? ∨ ?)
¬?
∴ ? ∨ ?
(b)
? → ? ∨ (¬?)
? → ? ∧ ?
∴ ? → ?
6. Prove that
(a) 9?
2 + 3? − 2 is even for any integer
n .
(b) For all integers m an n, m + n
and
m − n
are either both odd or both even.
(c) There are real numbers such that √? + ? = √? + √?.
(d) For all integers, if n is odd then ?
2
is odd.
7. Show that the following statements are false:
(a) There is an integer n such that 2?
2 − 5? + 2 is a prime.
(b) If m and n are any two positive integers then ?? > ? + ?.
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