Q.Write the converse, inverse, and contrapositive of each of the following implication.

Mathematical logic studies valid reasoning using precise mathematical symbols in place of ordinary language, so that whether an argument is correct can be checked mechanically rather than by intuition -- the same idea that lets a computer verify a program or an electrical circuit designed from $0$s and $1$s.

Statement (proposition). Definition 12.7: a declarative sentence that is either true or false, but never both, is called a statement or proposition. Imperative ("Give me that book"), exclamatory ("How beautiful!"), and interrogative ("Where are you going?") sentences are never propositions. An open sentence, such as "$x\times7=35$" or "He is a bad person", has a truth value that varies with an unstated condition (the value of $x$, or opinion) -- it only becomes a genuine proposition once that condition is pinned down (e.g. by a universal quantifier over a fixed domain). The truth value of a statement is $T$ (or $1$) if true, $F$ (or $0$) if false.

Simple vs compound statements. A statement that cannot be split into smaller statements is a simple (atomic) statement; one built by joining two or more simple statements with a connective is a compound (molecular) statement. Simple statements are named with propositional variables $p,q,r,\ldots$

Logical connectives join simple statements into compound ones. The five basic connectives, with their truth tables:

• Negation, $\neg p$ ("not $p$"): flips the truth value -- $\neg p$ is $T$ exactly when $p$ is $F$.
• Conjunction, $p\wedge q$ ("$p$ and $q$"): $T$ only when both $p$ and $q$ are $T$.
• Disjunction, $p\vee q$ ("$p$ or $q$"): $F$ only when both $p$ and $q$ are $F$ (otherwise $T$) -- this is the inclusive or.
• Conditional, $p\to q$ ("if $p$ then $q$", $p$ = antecedent/hypothesis, $q$ = consequent/conclusion): $F$ only when $p$ is $T$ and $q$ is $F$; otherwise $T$. In particular a conditional with a false hypothesis is automatically $T$ (vacuous truth), and $p\to q$ is judged purely by the symbols, never by whether $p$ and $q$ are related in meaning. …