Question

homework: 3.3 question 18, 3.3.41 part 3 of 8 construct a truth table for the given compound statement. ~(p ∧ q) ∨ ~r complete the truth table. p q r ~r p ∧ q ~(p ∧ q) ~(p ∧ q) ∨ ~r t t t f t f f t t f t t f t t f t

Explanation:

Step1: Determine negation of r

When \(r = T\), \(
eg r=F\); when \(r = F\), \(
eg r = T\).

Step2: Calculate \(p\land q\)

\(p\land q\) is \(T\) only when both \(p\) and \(q\) are \(T\), otherwise it is \(F\).

Step3: Find \(

eg(p\land q)\)
Negate the values of \(p\land q\). If \(p\land q=T\), then \(
eg(p\land q)=F\); if \(p\land q = F\), then \(
eg(p\land q)=T\).

Step4: Calculate \(

eg(p\land q)\lor
eg r\)
\(
eg(p\land q)\lor
eg r\) is \(T\) if either \(
eg(p\land q)\) or \(
eg r\) (or both) is \(T\), and \(F\) only when both \(
eg(p\land q)\) and \(
eg r\) are \(F\).

\(p\)

\(q\)

\(r\)

\(

eg r\) | \(p\land q\) | \(
eg(p\land q)\) | \(
eg(p\land q)\lor
eg r\) |

\(T\)	\(T\)	\(T\)	\(F\)	\(T\)	\(F\)	\(F\)
\(T\)	\(T\)	\(F\)	\(T\)	\(T\)	\(F\)	\(T\)
\(T\)	\(F\)	\(T\)	\(F\)	\(F\)	\(T\)	\(T\)
\(T\)	\(F\)	\(F\)	\(T\)	\(F\)	\(T\)	\(T\)
\(F\)	\(T\)	\(T\)	\(F\)	\(F\)	\(T\)	\(T\)
\(F\)	\(T\)	\(F\)	\(T\)	\(F\)	\(T\)	\(T\)
\(F\)	\(F\)	\(T\)	\(F\)	\(F\)	\(T\)	\(T\)
\(F\)	\(F\)	\(F\)	\(T\)	\(F\)	\(T\)	\(T\)

Answer:

\(p\)

\(q\)

\(r\)

\(

eg r\) | \(p\land q\) | \(
eg(p\land q)\) | \(
eg(p\land q)\lor
eg r\) |

\(T\)	\(T\)	\(T\)	\(F\)	\(T\)	\(F\)	\(F\)
\(T\)	\(T\)	\(F\)	\(T\)	\(T\)	\(F\)	\(T\)
\(T\)	\(F\)	\(T\)	\(F\)	\(F\)	\(T\)	\(T\)
\(T\)	\(F\)	\(F\)	\(T\)	\(F\)	\(T\)	\(T\)
\(F\)	\(T\)	\(T\)	\(F\)	\(F\)	\(T\)	\(T\)
\(F\)	\(T\)	\(F\)	\(T\)	\(F\)	\(T\)	\(T\)
\(F\)	\(F\)	\(T\)	\(F\)	\(F\)	\(T\)	\(T\)
\(F\)	\(F\)	\(F\)	\(T\)	\(F\)	\(T\)	\(T\)