1 - Sets

Set

A well-defined collection of distinct objects, considered as an object in its own right.

Empty Set

A set that contains no elements, denoted by the symbol φ or { }.

Finite Set

A set that contains a definite number of elements.

Infinite Set

A set that does not contain a definite number of elements.

Equal Sets

Two sets that contain exactly the same elements.

Subset

A set A is a subset of a set B if every element of A is also an element of B.

Proper Subset

A subset A of B that is not equal to B; that is, A ⊂ B and A ≠ B.

Universal Set

The set that contains all the objects under consideration for a particular discussion.

Union of Sets (A ∪ B)

The set of all elements that are in A, in B, or in both.

Intersection of Sets (A ∩ B)

The set of all elements that are common to both A and B.

Difference of Sets (A - B)

The set of elements that are in A but not in B.

Complement of a Set

The set of all elements in the universal set that are not in the given set.

Venn Diagram

A diagram that shows all possible logical relations between a finite collection of sets.

Roster Form

A way of representing a set by listing its elements within braces, such as {1, 2, 3}.

Set-builder Form

A method of specifying a set by stating a property that its members must satisfy.

Distinct Elements

Elements within a set that are not repeated; each is unique.

De Morgan's Laws

Rules that relate the complement of unions and intersections: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.

Well-defined Collection

A collection of objects for which it is possible to determine whether an object belongs or does not belong to the collection.

Natural Numbers (N)

The set of positive integers typically starting from 1: {1, 2, 3, ...}.

Rational Numbers (Q)

Numbers that can be expressed as the quotient of two integers, where the denominator is not zero.

Real Numbers (R)

All the numbers on the number line, including rational and irrational numbers.

Parameters of Sets

The characteristics that define elements of sets in mathematical expressions.