# [Discrete Mathematics] Logic and Proof

## I. Logic and Proof

#### 1. Propositional Logic

A proposition is a declarative sentence $that is, a sentence that declares a fact$ that is either true or false, but not both.

Name Meaning Notation
negation not p
disjunction p or q
conjunction p and q
conditional if p, then q
biconditional p if and only if q

#### 2. Propositional Equivalences

A compound proposition that is always true, no matter what the truth values of the propositional variables that occur in it, is called a tautology.

A compound proposition that is always false is called a contradiction.

A compound proposition that is neither a tautology nor a contradiction is called a contingency.

The compound propositions p and q are called logically equivalent if  is a tautology. 

Equivalence Name

Identity laws

Domination laws

Idempotent laws
Double negation law

Commutative laws

Associative laws

Distributive laws

De Morgan’s laws

Absorption laws

Negation laws
Table of common Logical equivalence

To be done

#### References

Discrete Mathematics and Its Applications 7th Edition, Kenneth H. Rosen

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