What is a statement?
A statement is a sentence that is either true or false — never both, never neither. “2 is even” is a statement (true). “The sequence 1/n converges” is a statement (true). “x + 1” is not a statement: it has no truth value until you say what x is. Analysis is built entirely out of statements, so the very first skill is recognizing which sentences carry a truth value and which are just expressions.
From simple statements we build compound ones using connectives. The four you need are and (P ∧ Q, true only when both hold), or (P ∨ Q, true when at least one holds — mathematicians always mean the inclusive “or”), not (¬P, flips the truth value), and implies (P ⇒ Q). The first three are intuitive. The fourth deserves its own section.
If P then Q
Implication P ⇒ Q is the workhorse of every theorem. It is false in exactly one situation: when P is true but Q is false. In all three other rows it is true. The surprising part is that P ⇒ Q is true whenever P is false — we call this vacuously true. “If 3 is even, then I am the moon” is a true statement, because its hypothesis never fires. A theorem only promises something when its hypothesis holds.
Truth table for P => Q P Q P => Q true true true true false false <-- the ONLY false row false true true (vacuously true) false false true (vacuously true) Read it as a promise: "if P happens, I guarantee Q." The promise is broken only when P happens and Q fails. If P never happens, the promise was never tested -> still kept. Example: "If a_n -> L and a_n -> M, then L = M" (limits are unique). When a sequence has no limit, the hypothesis P is false, so the statement holds vacuously for that sequence.
Three relatives travel with every implication. The [[converse|converse]] of P ⇒ Q is Q ⇒ P — a genuinely different statement that may be true or false on its own. The [[contrapositive|contrapositive]] is ¬Q ⇒ ¬P, which is logically equivalent to the original: proving one proves the other. And the inverse ¬P ⇒ ¬Q is equivalent to the converse. Confusing a statement with its converse is the single most common beginner error in proofs.
If and only if
When both P ⇒ Q and Q ⇒ P hold we write P ⇔ Q, read “P if and only if Q” (often abbreviated *iff*). Then P and Q are necessary and sufficient for each other: P is sufficient for Q (P guarantees Q) and necessary for Q (Q cannot hold without P). Definitions in analysis are secretly iff statements — “a sequence is bounded iff some M bounds all its terms.”