Question

complete the following truth table. use t for true and f for false. you may add more columns, but those added columns will not be graded.
p $q$ $(q \
ightarrow p) \wedge \sim q$
t t f
t f t
f t f
f f f
not graded

Explanation:

Step1: Define $\sim q$ values

For each row, $\sim q$ is the negation of $q$:

• Row1: $\sim T = F$
• Row2: $\sim F = T$
• Row3: $\sim T = F$
• Row4: $\sim F = T$

Step2: Calculate $q

ightarrow p$
Implication rule: $q
ightarrow p$ is F only if $q=T, p=F$:

• Row1: $T

ightarrow T = T$

• Row2: $F

ightarrow T = T$

• Row3: $T

ightarrow F = F$

• Row4: $F

ightarrow F = T$

Step3: Compute final conjunction

$\land$ is T only if both terms are T:

• Row1: $T \land F = F$
• Row2: $T \land T = T$
• Row3: $F \land F = F$
• Row4: $T \land T = F$

Step4: Extract required column

The missing column is $\sim q$, with values: F, T, F, T

Answer:

$p$

$q$

$\sim q$

$(q

ightarrow p) \land \sim q$ |

T	T	F	F
T	F	T	T
F	T	F	F
F	F	T	F