Unit 13: Mathematical Logic

1. Introduction to Logic and Propositions
2. Logical Connectives and Truth Tables
3. Conditional and Biconditional Statements
4. Tautologies, Contradictions, and Contingencies
5. Converse, Inverse, and Contrapositive
6. Exam Focus Enhancements

1. Introduction to Logic and Propositions

Mathematical Logic is the study of valid reasoning. It provides the rules for determining whether a mathematical statement is true or false.

What is a Proposition?

A Proposition (or statement) is a declarative sentence that is either True (T) or False (F), but not both.

Examples: "The sun rises in the east" (True), "10 + 5 = 20" (False).
Non-propositions: "How are you?" (Question), "Close the door" (Command), "x + 2 = 5" (Truth depends on x).

2. Logical Connectives and Truth Tables

Compound propositions are formed by combining existing propositions using Logical Connectives.

Operator	Name	Symbol	Condition for Truth
Negation	NOT	¬ p	Opposite of the original truth value.
Conjunction	AND	p \land q	True only if both are True.
Disjunction	OR	p \lor q	True if at least one is True.

3. Conditional and Biconditional Statements

Conditional Statement (p \to q)

Also known as an "If-Then" statement. It is false only when the hypothesis (p) is True and the conclusion (q) is False.

Biconditional Statement (p ≤ftrightarrow q)

Also known as "If and only if." It is true only when both p and q have the same truth values.

4. Tautologies, Contradictions, and Contingencies

By constructing a truth table for a compound proposition, we can classify it into one of three categories:

Tautology: A statement that is always True regardless of the truth values of its components (e.g., p \lor ¬ p).
Contradiction: A statement that is always False (e.g., p \land ¬ p).
Contingency: A statement that is neither a tautology nor a contradiction (it can be true or false).

5. Converse, Inverse, and Contrapositive

Given a conditional statement p \to q, we can derive three related statements:

Converse: q \to p
Inverse: ¬ p \to ¬ q
Contrapositive: ¬ q \to ¬ p

Exam Fact: A statement and its Contrapositive are logically equivalent (they always have the same truth value).

6. Exam Focus Enhancements

Exam Tips

Truth Table Rows: If there are n propositions, the truth table will have 2ⁿ rows. For two variables (p, q), use 4 rows.
Negation of Negation: Remember that ¬(¬ p) is equivalent to p.
Implication Shortcut: Think of p \to q as a promise. The only time the promise is broken is if p happens but q doesn't.

Common Mistakes

Non-Statements: Trying to identify truth values for commands or questions. They are not propositions!
OR vs XOR: In math, "OR" (\lor) is inclusive. It is true if both are true. Don't confuse it with "Either-Or" (XOR).
Inverse vs Converse: Getting these swapped. Remember: Contrapositive is the only one that is definitely equivalent to the original statement.

Frequently Asked Questions

Q: What is a Truth Value?
A: It is the attribute of a proposition being either True (represented as T or 1) or False (F or 0).

Q: Can a mathematical equation be a proposition?
A: Yes, if it is an equality or inequality without variables (e.g., 5 > 3 is a True proposition). Equations with variables are "Open Sentences."