Show that each of these conditional statements is a tautology by using truth tables. a) [¬p ∧ (p ∨ q)] → q b) [(p → q) ∧ (q → r)
] → (p → r) c) [p ∧ (p → q)] → q d) [(p ∨ q) ∧ (p → r) ∧ (q → r)] → r
2 answers:
Step-by-step explanation:
To show that a statement is a tautology using truth table - is to show that all the entries in the expression are truths T.
We can do this by taking each statement, expression by expression. For example, to show that
[~p ∧ (p ∨ q)] → q
is a tautology, knowing we have 3 columns, we have 2^3 = 8 rows. We start by putting putting truth values for p, q, and r respectively
Next, we find ~p, then find (p ∨ q), then find ~p ∧ (p ∨ q), before finally arriving at the required [~p ∧ (p ∨ q)] → q
TERMINOLOGIES AND SYMBOLS
- T means Truth
- F means False
- ~p means negation of p.
~p is F if p is T, and vice versa
- ∧ means conjunction.
p ∧ q is T only if p is T and q is T.
- ∨ means disjunction.
p ∨ q is T if either p or q is T.
- → is for conditional 'if then'
p → q is T if both p and q are T, or both p and q are F, or p is F and q is T, otherwise, it is F.
THE STEP BY STEP WORKINGS FOR THE STATEMENTS GIVEN ARE IN THE ATTACHMENT.
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