Logic

From AoPSWiki

Logic is the systematic use of symbolic and mathematical techniques to determine the forms of valid deductive or inductive argument.

Statements

A statement is either true or false, but it will never be both or neither. An example of statement can be "A duck is a bird." which is true. Another example is "A pencil does not exist" which is false.

Logical Notations

Main article: Logical notation

A Logical notation is a special syntax that is shorthand for logical statements.

Negations

A negation is denoted by $\neg p$ . $\neg p$ is the statement that is true when $p$ is false and the statement that is false when $p$ is true. This means simply "the opposite of $p$ "

Conjunction

The conjunction of two statements basically means " $p$ and $q$ " and is denoted by $p \land q$ .

Disjunction

The disjunction of two statements basically means " $p$ or $q$ " and is denoted by $p \land q$ .

Implication

This operation is given by the statement "If $p$ , then $q$ ". It is denoted by $p\Leftrightarrow q$ . An example is "if $x+3=5$ , then $x=2$ .

Converse

The converse of the statement $p \Leftrightarrow q$ is $q \Leftrightarrow p$ .

Contrapositive

The contrapositive of the statement $p \Leftrightarrow q$ is $\neg p \Leftrightarrow \neg q$

Truth Tables

A truth tale is the list of all possible values of a compound statement.

Quantifiers

There are two types of quantifiers: A universal Quantifier: "for all" and an existential Quantifier: "there exists". A universal quantifier is denoted by $\forall$ and an existential quantifier is denoted by $\exists$ .