Question

In which of the following cases is $(p\wedge q)\vee (\stackrel{}{\tilde{p}}\wedge \tilde{q})$ true?

• A. p is true, q is false.
• B. p is false, q is true.
• C. Both p and q are true.
• D. Both p and q are false.

Answer 1

Answer: (C), (D)

Both p and q are false.

Making the truth table
$$\begin{array}{ccccccc}\text{p}& \text{q}& p\wedge q& \tilde{p}& \tilde{q}& \tilde{p}\wedge \tilde{q}& (p\wedge q)\vee (p\wedge q)\\ \text{T}& \text{T}& \text{T}& \text{F}& \text{F}& \text{F}& \text{T}\\ \text{T}& \text{F}& \text{F}& \text{F}& \text{T}& \text{F}& \text{F}\\ \text{F}& \text{T}& \text{F}& \text{T}& \text{F}& \text{F}& \text{F}\\ \text{F}& \text{F}& \text{F}& \text{T}& \text{T}& \text{T}& \text{T}\end{array}$$
Hence it is true when both p and q are true or both are false.