Propositions and Compound Statements

Propositions and Compound Statements

1. Propositions

A proposition is a declarative sentence that is either true or false, but not both. It has a definite truth value.

Examples of propositions:

• "5 is an odd number." → True
• "The moon is made of cheese." → False

Not propositions:

• "What time is it?" (question)
• "Go to your room!" (command)
• "x + 2 = 7" (not a proposition until $x$ is given a value)

2. Truth Values

The truth value of a proposition is either:

• T (True)
• F (False)

3. Logical Connectives (Operators)

Propositions can be combined using logical operators to form compound statements.

a) Negation (¬p or ~p)

The negation of a proposition reverses its truth value.
If $p$ is true, then $\neg p$ is false.

b) Conjunction (p ∧ q)

The conjunction is true only if both $p$ and $q$ are true.

c) Disjunction (p ∨ q)

The disjunction is true if at least one of $p$ or $q$ is true.

d) Exclusive OR (p ⊕ q)

True only if exactly one of $p$, $q$ is true.

e) Implication (p → q)

"If $p$, then $q$"
The implication is false only when $p$ is true and $q$ is false.

f) Biconditional (p ↔ q)

"p if and only if q"
True when both have the same truth value.

4. Truth Tables

Truth tables help determine the truth value of compound statements for all possible truth values of their components.

Example: $(p \lor q) \land \neg p$

5. Logical Equivalence

Two statements are logically equivalent if they always have the same truth value. Denoted as $p \equiv q$

Examples:

• $p \to q \equiv \neg p \lor q$
• De Morgan’s Laws:
  • $\neg(p \land q) \equiv \neg p \lor \neg q$
  • $\neg(p \lor q) \equiv \neg p \land \neg q$

6. Tautology, Contradiction, and Contingency

• Tautology: A statement that is always true.
  Example: $p \lor \neg p$

• Contradiction: A statement that is always false.
  Example: $p \land \neg p$

• Contingency: A statement that is sometimes true and sometimes false depending on the truth values of its components.

Need the next topic?