Question

type the correct answer in each box. use t for true and f for false. complete the truth table for the inverse of a conditional statement. p q p→q ~p→~q t t t \boxed{} t f f \boxed{} f t t \boxed{} f f t \boxed{}

Explanation:

Step1: Define negations

For each row, find $\sim p$ (not p) and $\sim q$ (not q):

• Row1: $p=T, q=T$ → $\sim p=F, \sim q=F$
• Row2: $p=T, q=F$ → $\sim p=F, \sim q=T$
• Row3: $p=F, q=T$ → $\sim p=T, \sim q=F$
• Row4: $p=F, q=F$ → $\sim p=T, \sim q=T$

Step2: Evaluate $\sim p

ightarrow \sim q$
A conditional $a
ightarrow b$ is false **only** when $a=T$ and $b=F$; else true:

• Row1: $F

ightarrow F$ → $T$

• Row2: $F

ightarrow T$ → $T$

• Row3: $T

ightarrow F$ → $F$

• Row4: $T

ightarrow T$ → $T$

Answer:

Row1 (empty box): T
Row2 (empty box): T
Row3 (empty box): F
Row4 (empty box): T