Use the truth table to establish which of the following statements are tautologies and which are contradictions.

Question

(v)$(p\wedge(\sim p))\wedge((\sim q)\wedge p)$. This question is the fifth part of multipart q1

Answer 1

Toolbox:

• If $p$ and $q$ are two simple statements $p\wedge$ is the conjunction of $p$ and $q$ and $p \vee q$ is the disjunction .Negation of a statement $p$ is denoted by $\sim p$
• Rules for conjunction :
• $A_1$: The statement $p\wedge q$ has the truth table value $T$ whenever both $p$ and $q$ have the truth value $T$
• $A_2$: The statement $p\wedge q$ has the truth value $F$ whenever either $p$ or $q$ or both have the truth value $F$
• A statement is tautology if the last column in the truth table is $T$ entirely.It is a contradiction if the last column is $F$ entirely.

The statement is a contradiction.

$\begin{matrix} p & q & \sim p & \sim q & p \vee (\sim q) & (\sim q)\vee p & (p\wedge(\sim p))\wedge((\sim q)\wedge p) \\ T & T & F& F & F & F & F\\ T & F & F & T & F & T & F\\ F& T & T & F & F & F & F\\ F& F & T & T& F & F & F \end{matrix}$