The symbol '∧' represents the logical conjunction 'and'. It is used to combine two statements where both must be true for the entire expression to be true.

A conditional statement is represented by the implication arrow '→'. It expresses a relationship where, if the antecedent is true, then the consequent follows.

A conjunction 'p ∧ q' is true only if both p and q are true. Since q is false, the entire expression evaluates to false.

The inclusive 'or' is denoted by the symbol '∨', indicating that at least one of the propositions is true. This operator is fundamental in constructing compound logical expressions.

Negation is symbolized by '¬', which reverses the truth value of the proposition it precedes. This operation is essential in constructing and understanding logical statements.

Modus Ponens allows one to conclude 'q' from the premises 'if p then q' and 'p'. This inference is a foundational rule in logical reasoning and direct proofs.

The contrapositive of 'If p then q' is 'If not q then not p'. Therefore, 'If a number is not even, then it is not divisible by 4' is the correct equivalent formulation.

A direct proof begins by assuming the hypothesis and then logically deducing the conclusion. This step-by-step reasoning uses established facts and definitions to arrive at the result.

Proof by contradiction starts by assuming the opposite of what is to be proven and then showing that this assumption leads to an inconsistency. This contradiction confirms the validity of the original statement.

A biconditional statement 'p if and only if q' implies that p is both necessary and sufficient for q. This creates a two-way relationship, meaning both statements are true together or false together.

The fallacy 'post hoc ergo propter hoc' incorrectly assumes causation simply because one event follows another. Such reasoning does not provide valid evidence of a causal relationship.

Exclusive 'or' means that exactly one proposition is true, excluding the possibility that both are true simultaneously. This distinguishes it from the inclusive 'or' where both propositions can be true.

Modus Tollens permits the inference of 'not p' from the premises 'if p then q' and 'not q'. It is a fundamental rule of inference used in many logical proofs.

In classical logic, an implication is considered true if the antecedent is false. This holds irrespective of the truth value of the consequent.

To verify a universally quantified statement on a finite set, each element must be checked to ensure that P(x) holds true. This guarantees the statement's validity for the entire set.

A necessary condition is required for a conclusion to be true, whereas a sufficient condition, when met, guarantees the conclusion. This distinction is central to understanding conditional statements in logic.

Affirming the consequent is a fallacy that assumes the truth of the antecedent based solely on the truth of the consequent. This error reverses the logical flow of the original conditional statement.

Mathematical induction relies on proving a base case and then showing that if an arbitrary case holds, the subsequent case follows. This process confirms that the statement holds for all natural numbers.

A proof by cases requires that every possible scenario be examined and shown to lead to the same conclusion. This comprehensive approach ensures that no potential case is overlooked.

A counterexample offers a concrete case that shows a universal claim does not hold in all instances. This specific evidence invalidates the claim, contrasting with a failed proof which lacks sufficient support.