Many mathematical claims speak about how many objects satisfy a property. The words that carry this meaning are the quantifiers. There are two:
- The existential quantifier "there exists", symbol $\exists$ — asserts that at least one object has the property.
- The universal quantifier "for all" (or "for every"), symbol $\forall$ — asserts that every object has the property.
The first reads "there exists a real number $x$ such that $x^2 = 2$" (true). The second reads "for all real $x$, $x^2 \ge 0$" (also true). Quantifiers turn an open sentence into a genuine statement, because once the quantifier fixes the scope, the sentence has a definite truth value. A key skill is negating quantified statements: the negation of "for all" is "there exists … not", and the negation of "there exists" is "for all … not".
| Statement | Negation |
|---|---|
| $\forall\, x,\ P(x)$ | $\exists\, x : \sim P(x)$ |
| $\exists\, x : P(x)$ | $\forall\, x,\ \sim P(x)$ |
So the negation of "Every student passed" is "There exists a student who did not pass" — not "Every student failed". To disprove a "for all" claim you only need a single exception; to disprove a "there exists" claim you must rule out every case.
Implications: the "if-then" statement. The most important compound form in mathematics is the conditional "if $p$ then $q$", written $p \Rightarrow q$. Here $p$ is the hypothesis (or antecedent) and $q$ is the conclusion (or consequent). The conditional is false in exactly one case — when the hypothesis is true but the conclusion is false. In every other case it is true (including, perhaps surprisingly, whenever $p$ is false, which is called vacuously true).
| $p$ | $q$ | $p \Rightarrow q$ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
The same conditional $p \Rightarrow q$ can be phrased in several equivalent ways, and recognising them is essential:
- if $p$, then $q$
- $p$ implies $q$
- $p$ is sufficient for $q$
- $q$ is necessary for $p$
- $p$ only if $q$
- $q$ if $p$
Converse and contrapositive. From $p \Rightarrow q$ we build two related conditionals:
The contrapositive is logically equivalent to the original — it has the same truth value in every case, which is exactly why proving the contrapositive proves the original. The converse is not equivalent: a statement may be true while its converse is false. For "If a number is divisible by $6$, then it is divisible by $3$" (true), the converse "If a number is divisible by $3$, then it is divisible by $6$" is false ($9$ is a counter-example).
The biconditional: "if and only if". When both $p \Rightarrow q$ and its converse $q \Rightarrow p$ hold, we write $p \Leftrightarrow q$, read "$p$ if and only if $q$" (often abbreviated "iff"). It asserts that $p$ and $q$ are equivalent — each is both necessary and sufficient for the other. The biconditional is true exactly when $p$ and $q$ share the same truth value.
Deeper Insight — "necessary vs sufficient" and the contrapositive are the real prizes: Two ideas in this topic do far more work than the rest. First, the language of necessary and sufficient: in $p \Rightarrow q$, $p$ guarantees $q$ (sufficient) while $q$ is merely required by $p$ (necessary) — confusing the two is the same error as confusing a statement with its converse, and it is the most frequent mistake in proofs. Second, the equivalence of a conditional with its contrapositive: because $\sim q \Rightarrow \sim p$ always matches $p \Rightarrow q$, you can freely swap to whichever direction is easier to argue. This single equivalence is the engine behind the method of contrapositive in the next topic. Memorise that only the contrapositive (not the converse) preserves truth, and most reasoning questions become routine.