Q.Which one of the following statements has truth value $F$?

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. …