A set is a collection of any number of objects called members or elements. For example, a college faculty is the set of all the professors at the college, and the set of positive integers is the set of all integers greater than zero.

The symbol ∈ is used to specify that a particular element belongs to a set. a∈S means that a belongs to the set S.

Sets are usually represented by capital letters, but their elements can be specified in several different ways. Elements are usually represented by lower case letters.

Subsets

A set B is a subset of another set A if all of the elements in B are also elements of A:

B could contain one, two…or all of the elements that are in A. If B contains some, but not all, of the elements of A, it is called a proper subset of A. This is expressed with the notation:

Equal Sets

For sets A and B, A=B if and only if B ⊆A and A⊆B. The sets must have exactly the same elements, so each one is a
subset of the other. Neither is a proper subset of the other.

Sets as Elements

A set may contain another set as an element. For example, if B={1,2} and A={1,2,3{1,2}}, then B∈A. The elements 1
and 2 in A are different from the element {1,2} because {1,2} is a set, and 1 and 2 are not.

Special Sets

The null set is a set that has no elements, denoted ∅. It is defined to be a subset of every other set.

While individual sets may contain different groups belonging to some larger group, the universal set (U) is used to
specify the largest group it is necessary to represent. The whole it represents is different in different situations. For
example, if A={engineering students}, B={art students}, U may represent the set of all students. It could also represent
everyone affiliated with a college or university. The smaller sets (including ∅) are always subsets of U .

Try these exercises

Instructions

1.	Describe the set in words:
2.	Is either of these sets a subset of the other? Express the answer in proper notation.
3.	Which set is a subset of the other? Is it a proper subset?
4.	List all possible subsets of
5.	List all possible proper subsets of
6.	List all possible proper subsets of .
7.	Describe the set of even numbers that are also odd.
8.	S is the set of all integers greater than 2 and less than 15 that are either prime or multiples of 3. List the elements of S.
9.	A certain university requires all professors who teach there to have a doctorate in the subject they teach. T  is the set of all professors who teach at the university, and U  is the set of all professors who teach at the university and have a doctorate in the subject they teach. Is either U  or T  a subset of the other?
10.	What element do all sets have in common?

Answers to questions