Advanced Math

Q cards up to induction

Sets

A collection of objects called elements

Empty Set

Set that contains no elements

|A|= Cardinality of set A

If a set A has a finite number of elements, the … is the number of elements of A, without counting repetitive numbers

Subset

If A and B are sets, then we say that A is a…. of B if every element of A belongs to B.

Equal Sets

We say that A=B if and only if A and B have exactly the same elements. So, A is a subset of B and B is a subset of A

Fact

The nullset is a subset of every set

Relation

A relation R from A to B is any subset of AxB

Function

A relation R from A to B is a function if for every x in A there is one and only one y in B so that xRy

Injective

F is injective (one-to-one) if no two elements of A are mapped to the same element of B

Surjective

F is surjective (onto) if every elements of B is the image of an element of A

Bijective

F is bijective if it is both bijective and surjective

Partitions

Group sets into disjoint pieces.

Equivalence Relations

R is an equivalence relation if it is symmetric, reflexive and transitive

Fact

Any partition of a finite set is an equivalence relation.

Statement (proposition)

A statement is a sentence that is either true or false (but not both)

Logically Equivalent

Two statements are said to be logically equivalent if they have identical truth table values for all possible combinations of the variables

Tautology

A compound statement is a tautology if it is always true

Contradiction

A compound statement is a tautology if it is always false

Converse of “if p, then q”

“If q, then p”

Biconditional statement

Real numbers

Includes all numbers, including rational and irrational

Integers

Includes negatives, 0 and positive numbers

Natural numbers

Includes positive numbers only and could include 0

Rational numbers