MMW - Midterms

Sets

well-defined collection of objects.

Verbal description, Listing, & Set-builder notation

3 ways to denote a specific set

Empty Set

denoted ∅, if it does not contain any element.

Universal Set (Domain)

is the set that contains all objects of interest in a particular discourse.

Subset

A set B is said to be a _ of a set A if every element of B is also an element of A.

Relations

describes a connection between two or more sets

Reflexive Property

If x ∈ A, then x, x ∈ R. (“x is related to itself by R.”)

Symmetric Property

If x, y ∈ R, then y, x ∈ R.

Transitive Property

If x, y ∈ R and y, z ∈ R, then x, z ∈ R

Equivalence Relation

A relation R on a set A is said to be an _ if it satisfies the reflexive, symmetric, and transitive properties.

Negation ~ (“not”)

A statement which has the opposite truth value of another statement.

Conjunction ∧ (“and”)

Two or more statements joined using the word “and”

Disjunction ∨ (“or”)

Two or more statements joined using the word “or”

Implication → (“If... then...”)

Two statements joined using the word “If” and “then” and called a conditional statement 

Biconditional (“... if and only if ...”)

Two or more statements can be joined using the words “if and only if”

Truth table 

used to present all the possible truth values of one or more logical

statements.

Conjunction (AND) → p∧qp \land qp∧q

True only if both statements are true

Disjunction (OR) → p∨qp \lor qp∨q

True if at least one statement is true 

Negation (NOT) → ¬p\neg p¬p

Reverses the truth value of the statement

Implication (IF...THEN) → p→qp \rightarrow qp→q

False if only if p is true and q is flase

Biconditional (IF AND ONLY IF) → p↔qp \leftrightarrow qp↔q

True whe both statements have the same truth value