Section 6.1 Set and Set Operations

Set: A set is a collection of objects/elements. Ex. A = {w, a, r, d} Sets are often named with capital letters. Order of elements doesn’t matter, no duplicates. Set-builder notation: rule describes the definite property (properties) an object x must satisfy to be part of the set. Ex. B = {x | x is an even integer} Read: “x such that x is an even integer” Notation: w is an element of set A is written

Set Equality: Two sets A and B are equal, written A = B, if and only if they have exactly the same elements. Subset: If every element of a set A is also an element of a set B then A is a subset of B, and is written Ex. A = {w, a, r, d}; B = {d, r, a, w} Ex. A = {r, d}; B = {r, a, w, d, e, t} Every element in A is also in B Every element in A is in B and every element in B is in A.

Empty Set: The set that contains no elements is called the empty set and is denoted Universal Set: The set of all elements of interest in a particular discussion is called the universal set and is denoted U. Note: The empty set is a subset of every set

Set Union: Let A and B be sets. The union of A and B, written is the set of all elements that belong to either A or B. Set Intersection: Let A and B be sets. The intersection of A and B, written is the set of all elements that are common to A and B. Set Operations

Ex. Given the sets: Combine the sets Overlap of the sets

Venn Diagrams U AB – visual representation of sets Rectangle = Universal Set Sets are represented by circles

Venn Diagrams U AB C A C B U

Complement of a Set: If U is a universal set and A is a subset of U, then the set of all elements in U that are not in A is called the complement of A, written A C. Set Complementation

Set Operations Commutative Laws Associative Laws Distributive Laws

De Morgan’s Laws Let A and B be sets, then

Ex. Given the sets: Elements not in A. Elements in A and not in B.

Venn Diagrams U A AB U