1.1 Logic

Name

Symbol

Definition

Notations

Negation

¬

Not

¬p

Conjunction

∧

And

p∧q, p AND q

Disjunction

∨

Or

p ∨q, pOR q

Implication Conditional

→

if, then (if p implies q = if p then q)

p→q

Biconditional

↔

If and only if, if p then and only then q

p↔q

Converse

converse of p→q is q→p

q→p

Inverse

p→q, inverse: if not p then not q

¬p → ¬q

Contrapositive

p→q, contrapositive: if not q→not p

¬q → ¬p

Therefore

∴

Symbol indicating a conclusion

∴

Such that

:

"Such that" in set builder notation or logic

p:p=∅.

Universal Quantifier

∀

For All

∀p

Existential Quantifier

∃

There Exist

∃q

Existential Uniqueness Quantifier

∃!

There Exist Exactly One

∃!p

Logical Equivalence

≡

Equivalent to

p≡q

Contradiction

⊥

Contradiction

⊥

Tautology

⊤

Always True

⊤

End of Proof

∎, ⊣

End of Proof (last symbol of the proof).

∎.

Name

Equivalence

Identity Laws

p∧T≡p,p∨F≡p

Domination Laws, Null Law

p∨T≡T,p∧F≡F

Idempotent Laws

p∧p≡p,p∨p≡p

Double Negation Law

¬(¬p)≡p

Commutative Law

p∨q≡q∨p, p∧q≡q∧p

Associative Law

(p∨q)∨r≡p∨(q∨r), p∧(q∧r)≡(p∧q)∧r

Distributive Law

p∧(q∨r)≡(p∧q)∨(p∧r), p∨(q∧r)≡(p∨q)∧(p∨r)

De Morgan's Law

¬(p∧q)≡(¬p)∧(¬q), ¬(p∨q)≡(¬p)∨(¬q)

Absorption Law

p∧(p∨q)≡p, p∨(p∧q)≡p

Negation Law

p∧¬p≡F, p∨¬p≡T

Logical Equivalence

p↔q≡(p→q)∧(q→p)

Implication Law

p→q≡¬(p∨q)

Contrapositive Law

p→q≡¬p→¬q