1.1 Logic

Name

Symbol

Definition

Notations

Negation

$\neg$

Not

$\neg p$

Conjunction

$\wedge$

And

$p\wedge q$, $p$ AND $q$

Disjunction

$\vee$

Or

$p$ $\vee q$, $p$OR $q$

Implication Conditional

$\to$

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

$p\to q$

Biconditional

$\leftrightarrow$

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

$p\leftrightarrow q$

Converse

converse of $p\to q$ is $q\to p$

$q\to p$

Inverse

$p\to q$, inverse: if not $p$ then not $q$

$\neg p$ $\to$ $\neg q$

Contrapositive

$p\to q$, contrapositive: if not $q\to$not $p$

$\neg q$ $\to$ $\neg p$

Therefore

$\therefore$

Symbol indicating a conclusion

$\therefore$

Such that

$:$

"Such that" in set builder notation or logic

$p:p=\varnothing$.

Universal Quantifier

$\forall$

For All

$\forall p$

Existential Quantifier

$\exists$

There Exist

$\exists q$

Existential Uniqueness Quantifier

$\exists !$

There Exist Exactly One

$\exists !p$

Logical Equivalence

$\equiv$

Equivalent to

$p\equiv q$

Contradiction

$\perp$

Contradiction

$\perp$

Tautology

$\top$

Always True

$\top$

End of Proof

$∎$, $⊣$

End of Proof (last symbol of the proof).

$∎$.

Name

Equivalence

Identity Laws

$p\wedge T\equiv p,p\vee F\equiv p$

Domination Laws, Null Law

$p\vee T\equiv T,p\wedge F\equiv F$

Idempotent Laws

$p\wedge p\equiv p,p\vee p\equiv p$

Double Negation Law

$\neg (\neg p)\equiv p$

Commutative Law

$p\vee q\equiv q\vee p$, $p\wedge q\equiv q\wedge p$

Associative Law

$(p\vee q)\vee r\equiv p\vee (q\vee r),$ $p\wedge (q\wedge r)\equiv (p\wedge q)\wedge r$

Distributive Law

$p\wedge (q\vee r)\equiv (p\wedge q)\vee (p\wedge r)$, $p\vee (q\wedge r)\equiv (p\vee q)\wedge (p\vee r)$

De Morgan's Law

$\neg (p\wedge q)\equiv (\neg p)\wedge (\neg q)$, $\neg (p\vee q)\equiv (\neg p)\vee (\neg q)$

Absorption Law

$p\wedge (p\vee q)\equiv p$, $p\vee (p\wedge q)\equiv p$

Negation Law

$p\wedge \neg p\equiv F$, $p\vee \neg p\equiv T$

Logical Equivalence

$p\leftrightarrow q\equiv (p\to q)\wedge (q\to p)$

Implication Law

$p\to q\equiv \neg (p\vee q)$

Contrapositive Law

$p\to q\equiv \neg p\to \neg q$