The Timpson Lyceum · College Algebra

Lecture 2 (College Algebra)

Overview

This lecture completes propositional logic and builds up to first-order predicate logic, the language in which all of mathematics is written. It reviews the five basic logical operators and the equivalences that let us reword statements, then introduces predicates, quantifiers, and the precise rules for negating quantified statements. It closes with two staple proof strategies, the logical law of identity, and a first taste of axiom-based algebraic proofs. The goal is fluency: being able to read, negate, and prove statements built from propositions, predicates, and quantifiers.

Key Concepts

Proposition (statement). A sentence that is either true or false, but not both. "Proposition" and "statement" mean the same thing. Greek letters such as $\alpha, \beta$ stand in for arbitrary propositions.

The five basic operators. Each is defined by its truth table, which is its exact meaning — the English reading is only an aid to intuition.

Negation $\lnot\alpha$ — flips the truth value: true becomes false, false becomes true.
Conjunction $\alpha \land \beta$ — true only when both are true.
Disjunction $\alpha \lor \beta$ — the inclusive or: true when at least one is true (including both). In logic, $\lor$ is always inclusive. The exclusive or is a separate operator, $\oplus$ .
Implication $\alpha \to \beta$ — false only when $\alpha$ is true and $\beta$ is false; true in every other case. The left side is the antecedent, the right side the conclusion. When the antecedent can never be satisfied, the implication is vacuously true.
Biconditional / logical equivalence $\alpha \leftrightarrow \beta$ — true when $\alpha$ and $\beta$ have the same truth value. Saying " $\alpha$ if and only if $\beta$ " means $\alpha \to \beta$ and $\beta \to \alpha$ both hold (hence the two-way arrow).

Key equivalences.

De Morgan's laws: $\lnot(\alpha \land \beta) \equiv \lnot\alpha \lor \lnot\beta$ and $\lnot(\alpha \lor \beta) \equiv \lnot\alpha \land \lnot\beta$ . A negation distributed across an and yields an or, and across an or yields an and.
Contrapositive: $\alpha \to \beta \equiv \lnot\beta \to \lnot\alpha$ . An implication and its contrapositive are logically equivalent.
Double negation: $\lnot\lnot\alpha \equiv \alpha$ .

Predicate. A proposition-valued function of one or more variables: once you supply values for its variables, it becomes a proposition. For example $P(x): x > 2$ is not true or false until $x$ is given — $P(5)$ is true, $P(1)$ is false. A predicate of exactly two variables is called a relation (e.g. $x < y$ , often written in infix form rather than $\text{lessthan}(x,y)$ ). Every relation is a predicate, but not every predicate is a relation. Predicates can be combined with logical operators to build new predicates.

Quantifiers. Applying a quantifier to a predicate also turns it into a proposition.

Universal $\forall x\, P(x)$ — "for all $x$ , $P(x)$ ." True if and only if $P(x)$ is true for every $x$ ; otherwise false.
Existential $\exists x\, P(x)$ — "there exists an $x$ such that $P(x)$ ." Defined as $\lnot\,\forall x\,\lnot P(x)$ .

Order of quantifiers matters. $\forall x\,\exists y\,(x+y=0)$ is true (pick $y=-x$ ), but $\exists x\,\forall y\,(x+y=0)$ is false (no single $x$ works for every $y$ ). Swapping the two quantifiers changes the meaning.

Negating quantifiers. Pushing a negation through a quantifier flips it:

\begin{aligned} \lnot\,\forall x\, P(x) &\equiv \exists x\,\lnot P(x) \\ \lnot\,\exists x\, P(x) &\equiv \forall x\,\lnot P(x) \end{aligned}

So the negation of "all men in Georgia are at least six feet tall" is "there exists a man in Georgia who is less than six feet tall."

Two staple proof strategies (read off the implication truth table):

Modus ponens: from $\alpha$ true and $\alpha \to \beta$ true, conclude $\beta$ true.
Modus tollens: from $\lnot\beta$ and $\alpha \to \beta$ , conclude $\lnot\alpha$ (this is just the contrapositive at work).

Law of identity. $\forall x\,(x = x)$ — everything is identical with itself. This is a theorem of logic (more basic than any field-specific axiom), used to justify operating on both sides of an equation.

Operating on both sides. If $a = b$ and the operation $a \circ c$ and $b \circ c$ are defined, then $a \circ c = b \circ c$ for any operation $\circ$ (add, multiply, etc.). The one forbidden operation is division by zero, because $a/0$ is undefined.

The two working axioms used in the closing proofs:

\begin{aligned} \textbf{Axiom 1:}\quad & a + b = a + c \implies b = c \\ \textbf{Axiom 2:}\quad & (a \cdot b = a \cdot c \text{ and } a \neq 0) \implies b = c \end{aligned}

Worked Example

Prove: if $3x + 4 = 19$ , then $x = 5$ . Begin by assuming the antecedent, then rewrite each side so the cancellation axioms apply.

\begin{aligned} 3x + 4 &= 19 &&\text{(given antecedent)} \\ 3x + 4 &= 15 + 4 &&\text{(rewrite } 19 \text{)} \\ 3x &= 15 &&\text{(Axiom 1: cancel } +4\text{)} \\ 3x &= 3 \cdot 5 &&\text{(rewrite } 15 \text{)} \\ x &= 5 &&\text{(Axiom 2: cancel } 3 \neq 0\text{)} \quad\blacksquare \end{aligned}

Summary

A proposition is true-or-false; the five operators ( $\lnot, \land, \lor, \to, \leftrightarrow$ ) are defined by their truth tables, and $\lor$ is always inclusive.
De Morgan's laws, the contrapositive, and double negation let you reword statements into equivalent, often easier-to-prove forms.
A predicate becomes a proposition only when you plug in values or apply a quantifier; a two-variable predicate is a relation.
For all/there-exists are duals: $\lnot\forall x\,P(x) \equiv \exists x\,\lnot P(x)$ and $\lnot\exists x\,P(x) \equiv \forall x\,\lnot P(x)$ , and the order of mixed quantifiers changes the meaning.
Modus ponens ( $\alpha$ , $\alpha\to\beta \Rightarrow \beta$ ) and modus tollens ( $\lnot\beta$ , $\alpha\to\beta \Rightarrow \lnot\alpha$ ) are the two basic proof moves.
The law of identity ( $x = x$ ) justifies doing the same defined operation to both sides of an equation — the sole exception being division by zero.