Discrete Mathematics 0.2

Disjunction

True if at least one of P or Q is true.

Biconditional

True if P and Q are both true or both false.

Converse of P → Q

Q → P (not logically equivalent to the original implication in general).

Contrapositive of P → Q

¬Q → ¬P (logically equivalent to the original implication).

Necessary

P is necessary for Q means Q → P.

Sufficient

P is sufficient for Q means P → Q.

Existential Quantifier

"There exists" or "there is".

Negation with Quantifiers

¬∀xP(x) is equivalent to ∃x¬P(x). ¬∃xP(x) is equivalent to ∀x¬P(x).

Implicit Quantifiers

Sometimes predicates are assumed to be universally quantified if no quantifier is explicitly stated.

∧

AND

∨

OR

¬

NOT

⇒

Implies

⇔

If and only if

∀

For all

∃

There exists

f(x)

Value of function f at x

∅

Empty set

Discrete Mathematics

Involves the study of mathematical structures that are fundamentally discrete rather than continuous.

Atomic Statement

A statement that cannot be divided into smaller statements.

Conjunction

True if both P and Q are true.

Implication

True if P is false or Q is true, or both.

Negation

True if P is false.

Predicate

A statement with variables that becomes a statement when specific values are substituted.

Universal Quantifier

"For all" or "every".

∈

Element of

∉

Not an element of

Statement

A declarative sentence that is either true or false.

Molecular Statement

A statement that can be divided into smaller statements.