During AMC testing, the AoPS Wiki is in read-only mode. No edits can be made.

Difference between revisions of "Logic"

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 $q$"

Implication

This operation is given by the statement "If $p$, then $q$". It is denoted by $p\Leftrightarrow q$

Converse

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

Contrapositive

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

Quantifiers

There are two types of quantifiers: $\dot$ (Error compiling LaTeX. ! Extra }, or forgotten $.) Universal Quantifier: "for all"$\dot$(Error compiling LaTeX. ! Extra }, or forgotten$.) Existential Quantifier: "there exists"