The Timpson Lyceum · College Algebra

Lecture 1 (College Algebra)

Overview

This lecture opens the course with propositional logic, the foundation on which all of mathematics is built. Because mathematics is constructed entirely out of logic, we begin by isolating the parts of language that are perfectly unambiguous and learning to manipulate them precisely. We define a statement, build new statements from old ones using the connectives and, or, not, implies, and if and only if, and capture each connective's behavior in a truth table. We use these tools to negate compound statements (De Morgan's laws), to distinguish a conditional from its converse and contrapositive, and to define logical equivalence. By the end you can determine the truth value of any compound statement from the truth values of its parts.

Key Concepts

Three kinds of reasoning.

• Deductive — if the premises are true, the conclusion must be true. Its truth depends only on the form of the argument, not on facts about the world: "All A are B; c is an A; therefore c is a B." This is the only reasoning used in mathematics.
• Inductive — generalizing from many observed instances to a universal claim (the basis of the empirical sciences). Only probable, never certain.
• Abductive — inferring the best explanation for a particular observation. The least precise of the three.

Statement (proposition). A sentence that is either true or false, but not both. "Statement" and "proposition" mean the same thing; we usually denote statements by Greek letters $\alpha, \beta$. Examples: "$2 \times 2 = 5$" is a (false) statement; "Go home" and "$x + 2 = 5$" are not statements — the latter is a predicate whose truth depends on $x$, and it only becomes a statement once a quantifier is added.

Sentential connectives. Words/phrases that build new sentences from existing ones. We study only the truth-functional ones: the truth value of the result is fully determined by the truth values of its parts. (Words like "but," "since," "Jack believes…," and "it is probable that…" are not truth-functional and are excluded.) The five we use:

• Negation $\lnot\alpha$ ("not $\alpha$" / "$\alpha$ is false"). True exactly when $\alpha$ is false.
• Conjunction $\alpha \land \beta$ ("$\alpha$ and $\beta$"). True only when both $\alpha$ and $\beta$ are true.
• Disjunction $\alpha \lor \beta$ ("$\alpha$ or $\beta$"). False only when both are false; otherwise true. This is the inclusive or — it includes the case where both hold. The exclusive or $\alpha \oplus \beta$ (true when exactly one holds) exists but is rarely used.
• Implication $\alpha \to \beta$ ("if $\alpha$ then $\beta$"). False only when $\alpha$ is true and $\beta$ is false; otherwise true. $\alpha$ is the antecedent (condition), $\beta$ the conclusion.
• Biconditional / logical equivalence $\alpha \leftrightarrow \beta$ ("$\alpha$ if and only if $\beta$"). True exactly when $\alpha$ and $\beta$ have the same truth value.

Truth tables. Two statements give $2^2 = 4$ rows of possible truth combinations; the table records the result of the connective in each row.

Vacuous truth. When the antecedent of an implication is false, the implication is automatically true (there is no case to violate). E.g. "If $\alpha$ is true then $\beta$ is true" holds vacuously whenever $\alpha$ cannot be satisfied.

Phrasings of $\alpha \to \beta$. All of these mean the same thing: "if $\alpha$ then $\beta$," "$\beta$ if $\alpha$," "$\alpha$ only if $\beta$," "$\alpha$ is sufficient for $\beta$," "$\beta$ is necessary for $\alpha$." Note that "$\alpha$ only if $\beta$" is $\alpha \to \beta$ (not $\beta \to \alpha$): there is fire only if there is oxygen means fire $\to$ oxygen.

Contradictory vs. contrary. $\alpha$ and $\lnot\alpha$ are contradictory — exactly one is always true. "All swans are white" and "some swans are not white" are contradictory (correct negation). "All swans are white" and "all swans are black" are merely contrary — both can be false (e.g. all swans yellow).

Negating quantifiers. $\lnot(\text{for all } x,\ P(x))$ becomes "there exists $x$ such that $\lnot P(x)$," and $\lnot(\text{there exists } x,\ P(x))$ becomes "for all $x,\ \lnot P(x)$." So the negation of "$x + 2 = 5$ holds for all $x$" is "there exists $x$ with $x + 2 \ne 5$."

De Morgan's laws. Negation distributes across $\land$ and $\lor$ by flipping the connective:

• $\lnot(\alpha \land \beta) \equiv \lnot\alpha \lor \lnot\beta$
• $\lnot(\alpha \lor \beta) \equiv \lnot\alpha \land \lnot\beta$

Converse and contrapositive of $\alpha \to \beta$.

• Converse $\beta \to \alpha$ — not logically equivalent to the original.
• Contrapositive $\lnot\beta \to \lnot\alpha$ — logically equivalent to the original (verified by truth table).

For example, $x = 3 \to x^2 = 9$ is true, but its converse $x^2 = 9 \to x = 3$ is false (since $x = -3$ also gives $9$); its contrapositive $x^2 \ne 9 \to x \ne 3$ is true.

Proof, at a high level. Every theorem has the form $\alpha \to \beta$. Mathematics never asserts the antecedent itself — not even axioms are called "true"; we only assert that the axioms imply their theorems. This is why mathematical truth is independent of the physical world.

Order of precedence (loosest binds last, like $=$ in arithmetic): $\lnot$, then $\land$, then $\lor$, then $\to$, then $\leftrightarrow$. So $\alpha \lor \beta \land \lnot\alpha$ parses as $\alpha \lor (\beta \land \lnot\alpha)$.

Worked Example

Prove De Morgan's law $\lnot(\alpha \land \beta) \equiv \lnot\alpha \lor \lnot\beta$ by building both columns of a truth table and checking they match in every row.

The two expressions produce identical outputs (F, T, T, T) for every combination of inputs, so they are logically equivalent. Concretely, the negation of "today is Monday and the weather is cold" is "today is not Monday or the weather is not cold."

Summary

• A statement is a sentence that is exactly one of true or false; mathematics works only with statements and with deductive reasoning.
• The five truth-functional connectives are not $\lnot$, and $\land$, or $\lor$ (inclusive), implies $\to$, and if and only if $\leftrightarrow$; each is fully defined by its truth table.
• An implication $\alpha \to \beta$ is false only when $\alpha$ is true and $\beta$ is false — so it is vacuously true whenever the antecedent is false.
• De Morgan's laws push a negation through a connective and flip it: $\lnot(\alpha\land\beta)\equiv\lnot\alpha\lor\lnot\beta$ and $\lnot(\alpha\lor\beta)\equiv\lnot\alpha\land\lnot\beta$.
• The contrapositive $\lnot\beta\to\lnot\alpha$ is logically equivalent to $\alpha\to\beta$, but the converse $\beta\to\alpha$ is not.
• Logical equivalence means two statements share the same truth value in every row; precedence runs $\lnot,\ \land,\ \lor,\ \to,\ \leftrightarrow$.