Q.Write each of the following sentences in symbolic form using statement variables $p$ and $q$.

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.
• Biconditional, $p\leftrightarrow q$ ("$p$ if and only if $q$"): $T$ exactly when $p$ and $q$ share the same truth value.

A sixth connective, Exclusive OR (EOR), $p\veebar q$, is $T$ exactly when precisely one of $p,q$ is $T$ (not both) -- the "either...or...but not both" reading.

Derived conditionals. From $p\to q$, three related statements are built:

• Converse: $q\to p$
• Inverse: $\neg p\to\neg q$
• Contrapositive: $\neg q\to\neg p$

(Note the inverse and converse are each other's contrapositive, and -- as the Logical Equivalence concept shows -- the contrapositive is always logically equivalent to the original conditional, while the converse and inverse are not.)

Statement formulas and row count. An expression built from statements and connectives is a statement formula. A formula in $n$ distinct propositional variables has a truth table with exactly $2^n$ rows -- $2$ rows for one variable, $4$ for two, $8$ for three, and so on -- since each variable independently ranges over $T$ and $F$.