Propositions

A proposition has a truth value; logical connectives build compound statements.

Different ways to understand it

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Direct proof
A direct proof derives the conclusion from assumptions using justified logical steps.
Logical implication
An implication is false only when its premise is true and conclusion false.
Negation and De Morgan laws
Negation reverses truth values and exchanges universal with existential quantification.
Quantifiers
Universal and existential quantifiers specify how a statement ranges over a domain.
Sets
A set is a collection of distinct objects; membership is different from inclusion.