Question

How many rows appear in a truth table for each of these compound propositions?

a)\(p \to \neg p\)

b)$\mathbf{(}\mathit{p}\mathbf{\vee }\mathbf{\neg }\mathit{r}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathit{q}\mathbf{\vee }\mathbf{\neg }\mathit{s}\mathbf{)}$

c)$\mathit{q}\mathbf{\vee }\mathit{p}\mathbf{\vee }\mathbf{\neg }\mathit{s}\mathbf{\vee }\mathbf{\neg }\mathit{r}\mathbf{\vee }\mathbf{\neg }\mathit{t}\mathbf{\vee }\mathit{u}$

d) \((p \wedge r \wedge t) \leftrightarrow (q \wedge t)\)

Short answer

a) 2 rows

b) 16 rows

c)64rows

d) 16 rows

Step-by-step solution

Definition of truth table

A truth table is a mathematical table which is used in logic

Rows in a)

The given statement$p\to \neg p$ has only one variable.

Hence, there are$2^1=2$ rows.

Rows in b)

The given statement $(p\vee \neg r)\wedge (q\vee \neg s)$has variables.

Hence, there are $2^4=16$rows.

Rows in c)

The given statement has $q\vee p\vee \neg s\vee \neg r\vee \neg t\vee u$variables.

Hence, there are$2^6=64$ rows

Rows in d)

The given statement \(p \wedge r \wedge t) \leftrightarrow (q \wedge t)\) has 4 variables.

Hence, there are $2^4=16$rows.