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Some Basic Definitions. A set is a collection of objects.The objects that make up the set are called its elements or members.There is only one set that contains no elements and it is called the empty set or null set or void set. It is denoted by {} or
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1. Chapter 1: Sets Mathematical Proofs: A Transition to Advanced Mathematics, 2nd ed by Chartrand,Polimeni, and Zhang
3. Sets of Numbers N stands for the set of natural numbers:
{1,2,3,}
Z stands for the set of integers:
{0,1,2, ,3,}
R stands for the set of real numbers.
Q stands for the set of rational numbers:
{a/b | a,b ? R}
4. Subsets A set A is called a subset of a set B if every element of A also belongs to B. If A is a subset of B, then we write
Example:
Let A={1,2,3} and B={1,2,3,4,5,6,7} and C={2,4,6}.
Which sets are subsets?
5. If A, B and C are sets such that A is a subset of B and B is a subset of C, then A is a subset of C. Proof: Let A, B and C be sets such that A is a subset of B and B is a subset of C. Let x be an element of A.
Then, by definition of subset, x is an element of B.
Again, by definition of subset, x is an element of C.
Therefore, any element of A is an element of C and so A is a subset of C.
6. Definitions Two sets A and B are equal, denoted by A=B, if every element of A is in B and every element of B is in A. So A=B if and
A set A is a proper subset of B is A is a subset of B, but A is not equal to B. This is denoted
The set consisting of all subsets of a given set A is called the power set of A and is denoted by P(A).
7. Set Operations The union of two sets A and B is the set of all elements belonging to A or B.
AUB={x : x ? A or x ? B}
The intersection of two sets A and B is the set of all elements belonging to both A and B.
AnB={x: x?A and x?B}
