Q.Determine the truth value of each of the following statements.

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 $F$.