Answer:
See below for truth tables and proofs
Explanation:
Let's understand what a tautology is and what a contradiction is
statement which is always true is called a tautology. A statement which
is always false is called a contradiction.
A proposition p ⇒ q has a hypothesis p and a conclusion q
p ⇒ q is False
a) p ⇒ q is False if p is True and q is False. It is True for all other values of p and q
The truth table for p ⇒ q therefore is::
p q (p ⇒ q)
F F T
F T T
T F F
T T T
We have to prove that p AND q ⇒ p is a tautology for all p, q i.e. it is True no matter what the values of p and q are
p AND q ⇒ p can be re-written as (p AND q) ⇒ p since the AND operator has precedence over the implies operator
Here is the truth table for (p ∧ q) ⇒ p.
(∧ is the symbol for logical AND)
Here the hypothesis is (p ∧ q) and the conclusion is p.
It is False if and only if the hypothesis (p ∧ q) is T and p is False. That combination never occurs in the truth table
p q p ∧ q (p ∧ q) ⇒ p
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T T T T
T F F T
F T F T
F F F T
Since the expression (p ∧ q) ⇒ p is True for all values of p and q it is a tautology
b)
Truth Table for (p ∨ ¬p) where ¬p represents not p or p'
p ¬p (p ∧ ¬p)
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F T F
T F F
Since the expression is False for any p value, it is a contradiction