We think of sets as collections of objects. More formally, one takes the terms "set" and "element" as undefined terms.

Notation: For specifying sets, A = {a, b, c} or A = {a_n | a_n = 1/n, n = 2, 4, 6,…}
or A = {x Î R | -1 < x < 1}.

• Uppercase is used to denote set names. Lowercase is used to denote set members/elements.
• The ellipsis is used to extend an implicit rule for determining set members suggested by the first few terms enumerated.
• Explicit formation rules may also be used in place of ellipses.
• To say a is an element of A, we write a Î A.
• Some sets are so important, they are represented by special symbols: the set of natural numbers, N, the set of integers, Z, the set of real numbers, R. Note: in computer science N is understood to be {0, 1, 2, 3,…} while in mathematics N = Z⁺, the set of positive integers.

Axiom of Extension: A set is determined by its elements (completely.)

We assume that all sets are subsets of some big universal set, U, which changes depending on context.

Definition: A set A is a subset of a set B iff every element of A is an element of B. However, not every element of B has to be an element of A. To express the proposition that A is a subset of B write A Í B.

A Í B « " x, x Î A ® x Î B.
A Ë B « $ x | x Î A Ù x Ï B.   

Recall the negation of a conditional and negation of a quantified statement.

Example: Let A = {a, b, c} and B = {a, b, c, d}. Since a Î B, b Î B, and c Î B, A Í B.

Definition: Set A is equal to Set B iff A and B have exactly the same elements.

A = B « (A Í B) Ù (B Í A).

Example: Consider the sets A = {a, b, c} and B = {a, b, c, d}, C = {c, b, a}, D = {1, 2, 3}, E = {1, 2, 2}, and F = {3, 1, 2, 2}. Which of these sets are equal?

Solution: The sets A and C are equal because they contain exactly the same members, the characters a, b, and c. The fact that the order of the characters in A and C differs is insignificant as long as each member of A is a member of C and each member of C is a member of A. Similarly, D = F. The order does not matter and the fact that the number 2 appears twice in F is irrelevant.

Definition: A is a proper subset of B if A is a subset of B, but A is not equal to B. We write A Ì B.

So in the example above C and A are proper subsets of B, E is a proper subset of D and F.

Definition: Let A be a set with a finite number of elements. The cardinality of A, denoted by |A| or #A, is equal to the number of elements in A.

Consider sets A and B where A is a proper subset of B then

(A Ì B) ® (|A| < |B|).

Operations on Sets: There are 4 operations we consider:

1. Union
   A È B = {x | x Î A Ú x Î B}
2. Intersection
   A Ç B = {x | x Î A Ù x Î B}
3. Complement
   A^c = {x | x Î U Ù x Ï A}
   also written as U\A, U-A, A with a tilde or bar on top.
4. Difference
   B - A = {x | x Î B Ù x Ï A}
   Consider A = {a, b, c}, B = {a, b, c, d}; B - A = {d}.

Analogies: union = disjunction, intersection = conjunction.

Symmetric difference: A Å B = (A È B) - (A Ç B).

Draw Venn diagrams for union, intersection, complement, difference and symmetric difference.