Lecture 3 Handout
Example Proof 1
Prove that the set of all powers of 2 (beginning with 2) is a subset of the set of all even numbers
Given: Two sets A and B,
A  { a | a  2 j , where j  Z  }
B  {b | b  2  k , where k  Z  }
Statement
Let x be an arbitrary element of A. We can write x  2 j
Rewrite as x  2  2 j 1
Again rewrite as x  2  m, where m  Z 
xB
Since x is an arbitrary element,
all elements of A must be  B
AB
Justification
Given
Factoring
Substitution. 2 is an integer, 2 raised to an integer power is an integer
Definition of membership in B
Definition of inclusion
Definition of subset
Example Proof 2
Prove that for any two sets A and B that A  B  A  B
Statement
Let x be an arbitrary element of A B
x  A and x  B
Since x  A , x  A  B
Since x is an arbitrary element,
all elements of A B must be  A  B
AB AB
Justification
Given
Definition of intersection
Definition of union
Definition of inclusion
Definition of subset