Page 3 - 11-Math-2 Sets Functions and Groups

21.. SQeutsaFdurantcitcioEnqsuaantdioGnrsoups eeLLeeaarrnn..PPuunnjjaabb

O = The set of all odd integers = { ± 1,± 3,±5,...}.
E = The set of all even integers = {0,±2,±4,...}.

Q = The set of all rational numbers =  x = p 
Q’ = The set of all irrational numbers x q where p,q ∈ Z and q ≠0



=  x ≠p where p,q ∈ Z and q 
x q ≠0



= The set of all real numbers = Q ∪ Q’

Equal Sets: Two sets A and B are equal i.e., A=B, if and only if they have the same elements
that is, if and only if every element of each set is an element of the other set.
Thus the sets { 1, 2, 3 } and { 2, 1, 3} are equal. From the definition of equality of sets
it follows that a mere change in the order of the elements of a set does not alter the set. In
other words, while describing a set in the tabular form its elements may be written in any
order.

Note: (1) A = B if and only if they have the same elements means
if A = B they have the same elements and if A and B have the same elements
then A = B.

(2) The phrase if and only if is shortly written as “iff “.

Equivalent Sets: If the elements of two sets A and B can be paired in such a way that each element
of A is paired with one and only one element of B and vice versa, then such a pairing is called a
one-to-one correspondence between A and B e.g., if A = {Billal, Ahsan, Jehanzeb} and B = {Fatima,
Ummara, Samina} then six different (1 - 1) correspondences can be established between A and B

Two of these correspondences are shown below; -

i). {Billal, Ahsan, Jehanzeb }

 

{Fatima, Ummara, Samina }

version: 1.1

3