Question

complete a truth table to determine whether the negation of p→q is p∧~q
complete the truth table.
truth table with columns: p, q, ~q, p∧~q, p→q, ~(p→q); rows with t/f entries and blank cells to fill

Explanation:

Step1: Analyze $

eg q$ (row 3)
For $q = T$ (row 2), $
eg q = F$; for $q = F$ (row 3), $
eg q = T$; for $q = T$ (row 4), $
eg q = F$; for $q = F$ (row 5), $
eg q = T$. Wait, original table rows: row1: $p=T,q=T$; row2: $p=T,q=F$; row3: $p=F,q=T$; row4: $p=F,q=F$. So:

• Row3 ($p=F,q=T$): $

eg q = F$

• Row4 ($p=F,q=F$): $

eg q = T$

Step2: Analyze $p\land

eg q$ (row 4)

• Row3 ($p=F,

eg q=F$): $F\land F = F$

• Row4 ($p=F,

eg q=T$): $F\land T = F$

Step3: Analyze $p

ightarrow q$ (row 5)

• Row3 ($p=F,q=T$): $F

ightarrow T = T$

• Row4 ($p=F,q=F$): $F

ightarrow F = T$

Step4: Analyze $

eg(p
ightarrow q)$ (row 6)

• Row3 ($p

ightarrow q=T$): $
eg T = F$

• Row4 ($p

ightarrow q=T$): $
eg T = F$

Filling the table (rows 3 - 6, columns 3 - 4):

$p$

$q$

$

eg q$ | $p\land
eg q$ | $p
ightarrow q$ | $
eg(p
ightarrow q)$ |

$T$	$T$	$F$	$F$	$T$	$F$
$T$	$F$	$T$	$T$	$F$	$T$
$F$	$T$	$F$	$F$	$T$	$F$
$F$	$F$	$T$	$F$	$T$	$F$

Answer:

Filled truth table as above (missing cells: $
eg q$: F, F; $p\land
eg q$: F, F; $p
ightarrow q$: T, T; $
eg(p
ightarrow q)$: F, F for rows 3 - 4).