## SET THEORY 1/3

#### 1. SETS AND SUBSETS

A set is a clearly defined collection of objects. These objects are called the elements or members of the set. For example, the socks in a drawer form a set.The collection of all books in a library are a set; the prime numbers between 1 and 100 are a set. Sets are usually named with capital letters, while the elements of a set are designated by lower case letters. Here are some fundamental concepts of set theory and the notation used:

"The set A is equal to the set B."  Two sets are equal if they contain all of the same elements.
"The set A is a subset of the set B."  A set A is a subset of a set B if every element of A is also an element of B.
"a is an element of the set B."  This notation indicates that the element a is a member of the set B.
The term universal set, usually designated by U (or ), refers to the set of all elements under consideration in a particular discussion or group of exercises.

Another important set is the set that has no elements, known as the empty set or the null set.  It is designated with the symbol Æ, or empty set braces,  {  }, to indicate a set that contains no elements.

Consider the sets  A = {2, 4, 6, 8, 10, 12}    B = {4, 6}  and C = {1, 3, 5}

Then we could write 2ÎA,  5ÏA, BÌA, CË A ,  A Ì A (every set is a subset of itself) and  Æ ÌA (the null set is a subset of every set).

Below are some important sets of numbers that are undoubtedly familiar to you already!!

 - natural numbers - integer numbers - rational numbers - real numbers

Using the symbols from above, we can show the relationship between these sets: and also depict it with a diagram: