Q.Fill in the blank: If the truth value of a molecular statement is true, then every value of its constituent statement is false (F) ____.

Mathematical Logic

Mathematical logic studies statements and the connectives that combine them: negation $\sim p$, conjunction $p\wedge q$, disjunction $p\vee q$, conditional $p\to q$ and biconditional $p\leftrightarrow q$. Their truth values are fixed by truth tables; a statement true for all inputs is a tautology, one false for all is a contradiction.

Important equivalences:

• $p\to q\equiv\sim p\vee q$ (so $\sim(p\to q)\equiv p\wedge\sim q$).
• De Morgan: $\sim(p\wedge q)\equiv\sim p\vee\sim q$, $\sim(p\vee q)\equiv\sim p\wedge\sim q$.
• Contrapositive $\sim q\to\sim p\equiv p\to q$; the inverse is $\sim p\to\sim q$.
• Quantifier negation: $\sim(\forall x\,P)\equiv\exists x\,\sim P$.

'$p$ only if $q$' and '$q$ is necessary for $p$' both mean $p\to q$. Switching networks are modelled with $\wedge$ (series) and $\vee$ (parallel), and simplified using these laws to decide truth values or equivalent forms. …