Question

construct the truth - table for the following:

1. $(p\vee q)\wedge\

eg(p\wedge q)$

1. $(p\

ightarrow q)\longleftrightarrow(\
eg p\vee q)$

Explanation:

Step1: List all possible truth - value combinations

We have two propositional variables \(P\) and \(Q\). So, there are \(2^2 = 4\) possible combinations: \((T,T)\), \((T,F)\), \((F,T)\), \((F,F)\).

Step2: Calculate \(P\vee Q\) values

The disjunction \(P\vee Q\) is true if either \(P\) is true or \(Q\) is true or both are true.

• When \(P = T\) and \(Q = T\), \(P\vee Q=T\).
• When \(P = T\) and \(Q = F\), \(P\vee Q=T\).
• When \(P = F\) and \(Q = T\), \(P\vee Q=T\).
• When \(P = F\) and \(Q = F\), \(P\vee Q=F\).

Step3: Calculate \(P\wedge Q\) values

The conjunction \(P\wedge Q\) is true only when both \(P\) and \(Q\) are true.

• When \(P = T\) and \(Q = T\), \(P\wedge Q=T\).
• When \(P = T\) and \(Q = F\), \(P\wedge Q=F\).
• When \(P = F\) and \(Q = T\), \(P\wedge Q=F\).
• When \(P = F\) and \(Q = F\), \(P\wedge Q=F\).

Step4: Calculate \(

eg(P\wedge Q)\) values
Negation of \(P\wedge Q\).

• When \(P\wedge Q=T\), \(

eg(P\wedge Q)=F\).

• When \(P\wedge Q=F\), \(

eg(P\wedge Q)=T\).

Step5: Calculate \((P\vee Q)\wedge

eg(P\wedge Q)\) values
The conjunction of \(P\vee Q\) and \(
eg(P\wedge Q)\).

• When \(P\vee Q = T\) and \(

eg(P\wedge Q)=F\) (i.e., \(P = T\), \(Q = T\)), \((P\vee Q)\wedge
eg(P\wedge Q)=F\).

• When \(P\vee Q = T\) and \(

eg(P\wedge Q)=T\) (i.e., \(P = T\), \(Q = F\)), \((P\vee Q)\wedge
eg(P\wedge Q)=T\).

• When \(P\vee Q = T\) and \(

eg(P\wedge Q)=T\) (i.e., \(P = F\), \(Q = T\)), \((P\vee Q)\wedge
eg(P\wedge Q)=T\).

• When \(P\vee Q = F\) and \(

eg(P\wedge Q)=T\) (i.e., \(P = F\), \(Q = F\)), \((P\vee Q)\wedge
eg(P\wedge Q)=F\).

The truth - table is as follows:

\(P\)

\(Q\)

\(P\vee Q\)

\(P\wedge Q\)

\(

eg(P\wedge Q)\) | \((P\vee Q)\wedge
eg(P\wedge Q)\) |

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

Answer:

The truth - table for \((P\vee Q)\wedge
eg(P\wedge Q)\) is shown above.