QUESTION IMAGE
Question
warm-up/class practice!
- create a complete truth table for the following compound statement. tell whether the statement is a tautology, self-contradiction or neither.
$(sim p \
ightarrow q) \lor (q \land r)$
- use one of demorgan’s laws to rewrite the following statement:
a. it is not the case that you need a dated receipt and your credit card to return this item.
Problem 1
Step1: List all variable truth values
There are 3 variables ($p, q, r$), so $2^3=8$ truth value combinations:
| $p$ | $q$ | $r$ |
|---|---|---|
| T | T | F |
| T | F | T |
| T | F | F |
| F | T | T |
| F | T | F |
| F | F | T |
| F | F | F |
Step2: Calculate $\sim p$
Negate the truth values of $p$:
| $p$ | $q$ | $r$ | $\sim p$ |
|---|---|---|---|
| T | T | F | F |
| T | F | T | F |
| T | F | F | F |
| F | T | T | T |
| F | T | F | T |
| F | F | T | T |
| F | F | F | T |
Step3: Calculate $\sim p
ightarrow q$
Recall $a
ightarrow b$ is F only when $a=T, b=F$:
| $p$ | $q$ | $r$ | $\sim p$ | $\sim p |
ightarrow q$ |
| T | T | T | F | T |
| T | T | F | F | T |
| T | F | T | F | T |
| T | F | F | F | T |
| F | T | T | T | T |
| F | T | F | T | T |
| F | F | T | T | F |
| F | F | F | T | F |
Step4: Calculate $q \land r$
Recall $a \land b$ is T only when $a=T, b=T$:
| $p$ | $q$ | $r$ | $\sim p$ | $\sim p |
ightarrow q$ | $q \land r$ |
| T | T | T | F | T | T |
| T | T | F | F | T | F |
| T | F | T | F | T | F |
| T | F | F | F | T | F |
| F | T | T | T | T | T |
| F | T | F | T | T | F |
| F | F | T | T | F | F |
| F | F | F | T | F | F |
Step5: Calculate $(\sim p
ightarrow q) \lor (q \land r)$
Recall $a \lor b$ is T if at least one is T:
| $p$ | $q$ | $r$ | $\sim p$ | $\sim p |
ightarrow q$ | $q \land r$ | $(\sim p
ightarrow q) \lor (q \land r)$ |
| T | T | T | F | T | T | T |
| T | T | F | F | T | F | T |
| T | F | T | F | T | F | T |
| T | F | F | F | T | F | T |
| F | T | T | T | T | T | T |
| F | T | F | T | T | F | T |
| F | F | T | T | F | F | F |
| F | F | F | T | F | F | F |
Step6: Classify the statement
A tautology is always T, self-contradiction always F. This statement has both T and F values.
DeMorgan's Law states $\sim (a \land b) \equiv \sim a \lor \sim b$. Let $a$ = "you need a dated receipt" and $b$ = "you need your credit card". The original statement is $\sim(a \land b)$, so rewrite it using the law.
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The complete truth table is shown above. The statement is neither a tautology nor a self-contradiction.
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