1 Mathematical Statements, Operations on Statements, Truth Tables (Theory)

What is a mathematical statement?

(1) assigned a truth value

(2) classified as true or false, but not both

(3) represented in lowercase letters

What describes a non-mathematical statement?

(1) non declarative statements

(2) opinion

(3) questions

(4) exclamations

(5) imperatives

Conjunction (Notation and Description)

(1) p ∧ q

(2) "and"

A conjunction is true if...

both p and q are true

Disjunction (Notation and Description)

(1) p ∨ q

(2) "or"

A disjunction is true if...

at least one statement is true

Conditional (Notation and Description)

(1) p -> q

(2) "if, then"

A conditional is true if...

(1) premise = true, conclusion = true

(2) premise = false, conclusion = false

(3) premise = false, conclusion = true

NEVER (4) premise = true, conclusion = false

Biconditional (Notation and Description)

(1) p <‒> q

(2) if and only if

A biconditional statement is true if...

(1) premise = true, conclusion = true

(2) premise = false, conclusion = false

NEVER (3) F <-> T, or (4) T <-> F

Negation (Notation and Description)

(1) ~p

(2) "not"

A negated statement is true if...

(1) premise = false, or ~F, "not false"

What are truth tables?

(1) truth tables basically go through every single case of the problem

Formula to Determine All Cases

2^n

Tautology

(1) logically true statement that is true in all circumstances