The Timpson Lyceum · College Algebra

Lecture 3 (College Algebra)

Overview

This lecture builds algebra from the ground up: instead of taking the usual rules for granted, we treat algebra as logic applied to a small list of stated assumptions. We first review how predicates become propositions, then practice writing rigorous proofs of simple equations using only two cancellation axioms plus the rules of logic. The point is to separate what comes from logic (true in every field of mathematics) from what comes from algebra (assumed only here), and to translate everyday phrasings — "if and only if," "only if," "necessary," "sufficient" — into precise implications. The lecture then introduces the field axioms for a mystery set $\mathbb{R}$ of "numbers" and shows how the entire number line emerges from just two given elements, $0$ and $1$. By the end you should be able to prove a basic algebraic statement line by line, justifying every step.

Key Concepts

Proposition vs. predicate.

• A proposition is a sentence that is either true or false (never both).
• A predicate is a proposition-valued function: "$x > 2$" is not true or false until $x$ is filled in.
• Two ways to turn a predicate into a proposition: substitute a value for the variable, or attach a quantifier.
• Existential $\exists x\,P(x)$ ("there exists an $x$ such that…") and universal $\forall x\,P(x)$ ("for all $x$…") both produce propositions.

Logic vs. algebra. Logic sits beneath all of mathematics; you may always use it, in any field. The two cancellation axioms below are assumed only in algebra. Two related statements come purely from logic with identity (everything is identical to itself, $a = a$):

• $\forall a,b,c:\ a = b \implies a + c = b + c$
• $\forall a,b,c:\ a = b \implies a \cdot c = b \cdot c$

These let you "do the same thing to both sides" — no algebraic assumption needed.

The two cancellation axioms.

• Axiom 1 (additive cancellation): $\forall a,b,c:\ a + b = a + c \implies b = c.$
• Axiom 2 (multiplicative cancellation): $\forall a,b,c\ (a \neq 0):\ a\cdot b = a\cdot c \implies b = c.$ The restriction $a \neq 0$ matters: $0\cdot 2 = 0\cdot 3$ does not give $2 = 3$.

Translating English connectives. With antecedent $\alpha$ and consequent $\beta$:

• "$\alpha$ only if $\beta$" means $\alpha \implies \beta$.
• "$\alpha$ if $\beta$" means $\beta \implies \alpha$.
• "$\alpha$ if and only if $\beta$" (written $\alpha \iff \beta$) means $\alpha \implies \beta$ and $\beta \implies \alpha$.
• "$\beta$ is a necessary condition for $\alpha$" means $\alpha \implies \beta$ (e.g. oxygen is necessary for fire: $\text{fire} \implies \text{oxygen}$).
• "$\alpha$ is a sufficient condition for $\beta$" means $\alpha \implies \beta$ (fire is sufficient for oxygen).

Logical equivalences used in proofs.

• Contrapositive: $\alpha \implies \beta$ is equivalent to $\lnot\beta \implies \lnot\alpha$.
• Implication as disjunction: $\alpha \implies \beta$ is equivalent to $\lnot\alpha \lor \beta$.
• Negating quantifiers: $\lnot\exists x\,P(x) \equiv \forall x\,\lnot P(x)$ and $\lnot\forall x\,P(x) \equiv \exists x\,\lnot P(x)$.
• De Morgan: $\lnot(\alpha \land \beta) \equiv \lnot\alpha \lor \lnot\beta$.

Proof hygiene. Begin with Proof: and underline the statement. Use let to introduce/assume the antecedent, observe when invoking something already established. Every line should read as a complete sentence; finish with $\blacksquare$.

The field axioms (the mystery set $\mathbb{R}$). $\mathbb{R}$ is a set whose elements we call numbers; it is closed under two operations $+$ and $\cdot$ (a sum and product of numbers is again a number), satisfying:

• Commutativity: $x + y = y + x$ and $x\cdot y = y\cdot x$.
• Associativity: $(x + y) + z = x + (y + z)$ and $(x\cdot y)\cdot z = x\cdot(y\cdot z)$.
• Additive identity: there exists $0$ with $x + 0 = x$.
• Multiplicative identity: there exists $1$ with $x\cdot 1 = x$.
• Distinctness: $0 \neq 1$ (must be assumed — different symbols do not by themselves mean different numbers).
• Additive inverse: for each $x$ there exists $y$ with $x + y = 0$.
• Multiplicative inverse: for each $x \neq 0$ there exists $y$ with $x\cdot y = 1$.
• Distributivity: $x\cdot(y + z) = x\cdot y + x\cdot z$.

Building the numbers. From $0$ and $1$ alone, define $2 := 1 + 1$, $3 := 2 + 1$, and so on (the successor construction). Inverses give the negatives ($-1$ is the additive inverse of $1$) and the reciprocals ($\tfrac13$ is the multiplicative inverse of $3$). Undefined terms: $0$ and $1$ are not defined — like words in a dictionary you cannot define everything without circularity, so the axioms themselves give them meaning.

Worked Example

Prove: if $3x + 4 = 19$ then $x = 5$, using only the axioms and definitions of numbers.

State the tools first — additive cancellation $a + b = a + c \implies b = c$ (Axiom 1), multiplicative cancellation $a\cdot b = a\cdot c \implies b = c$ for $a \neq 0$ (Axiom 2) — then assume the antecedent and work down:

A second, more revealing example shows why the logic comes first. Prove: there is no number $x$ with $3x + 5 = 11$ and $4x - 1 = 3$. Negating the existential and pushing the negation through with De Morgan and "implication as disjunction" turns the goal into the much friendlier statement $\forall x:\ \big(3x + 5 = 11 \implies 4x - 1 \neq 3\big)$. Now take any such $x$:

The hard part was not the arithmetic but using logic to figure out what was actually being asked.

Summary

• A predicate becomes a proposition by substituting a value or by adding a quantifier ($\exists$ or $\forall$).
• Logic (including identity, $a = a$) underlies every field; the cancellation axioms are assumed only for algebra. "Do the same thing to both sides" is logic, not an axiom.
• Axiom 1 cancels a common additive term; Axiom 2 cancels a common multiplicative factor, but only when that factor is nonzero.
• Translate English carefully: "only if" is $\implies$ forward, "if" is $\implies$ backward, "iff" is both; "necessary" is the consequent, "sufficient" is the antecedent. Use the contrapositive $(\lnot\beta \implies \lnot\alpha)$ and De Morgan / quantifier-negation laws to reshape a goal into something provable.
• The reals form a field: closed under $+$ and $\cdot$, both commutative and associative, with identities $0$ and $1$ (assumed distinct), inverses, and distributivity tying the two operations together.
• Every familiar number is constructed: successors $2 = 1+1, 3 = 2+1, \dots$ from $0$ and $1$, negatives from additive inverses, fractions from multiplicative inverses — while $0$ and $1$ themselves stay undefined, given meaning only by the axioms.