Generalised Union and Intersection


Generalised Union

• The union of a collection of set is a 
• set that contains those elements that are members of at 
• least one set in the collection
  
  
•      A ∪ (B ∪ C) = (A ∪ B) ∪ C

• 
  A = {0,2,4,6}, B = {0,1,3,5,7}, C = {0,1,2,3}
• 
  
  A ∪ B ∪ C = {0,1,2,3,4,5,6,7}

• 
  

Generalised Intersection

• The intersection of a collection of sets is a set that 
• contains those elements that are members all the sets in 
• the collection
• 
  

•     A ∩ (B ∩ C) = (A ∩ B) ∩ C

• A = {0,2,4,6}, B = {0,1,3,5,7}, C = {0,1,2,3}
• 
  
  A ∩ B ∩ C = {0}

Cartesian Product

 The direct product of two sets

Specifically, the Cartesian product of two sets X and Y denoted  X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y

Example

A={abu,dol}

B={cow,goat,camel}

What is the cartesian product of A x B ?

  

       A x B ={(abu,cow),(abu,goat),(abu,camel),(dol,cow),(dol,goat),(dol,camel)}

Take note !!

      A x B  B x A

POWER SET

The power set of a set A is the set of all its subsets (including, of course, itself and the empty set). It is denoted by P(A).

Using set comprehension notation, P(A) can be defined as

P(A) = { Q | Q⊆A }

Example 4

Write down the power sets of A if:

(a) A = {1, 2, 3}

(b) A = {1, 2}

(c) A = {1}

(d) A = ø

Solution

(a) P(A) = { {1, 2, 3}, {2, 3}, {1, 3}, {1, 2}, {1}, {2}, {3}, ø }

(b) P(A) = { {1, 2}, {1}, {2}, ø }

(c) P(A) = { {1}, ø }

(d) P(A) = { ø }

SET OPERATION

·         Union

Definition (Union): The union of sets A and B, denoted by AB , is the set defined as :-
             AB = { x | xAxB }

Example 1: If A = {1, 2, 3} and B = {4, 5} ,  then AB = {1, 2, 3, 4, 5} .

Example 2: If A = {1, 2, 3} and B = {1, 2, 4, 5} ,  then AB = {1, 2, 3, 4, 5} .

Note that elements are not repeated in a set.

·         Intersection

Definition (Intersection): The intersection of sets A and B, denoted by AB , is the set defined as :-
             AB = { x | xAxB }

Example 3: If A = {1, 2, 3} and B = {1, 2, 4, 5} ,  then AB = {1, 2} .

Example 4: If A = {1, 2, 3} and B = {4, 5} ,  then AB = .

·         Disjoin Set  (Non-Overlapping Sets )

Two or more sets which have no elements in common. For example, the sets A = {a,b,c} and B = {d,e,f} are disjoint.

·         Set Difference

Definition (Difference): The difference of sets A from B , denoted by A- B , is the set defined as

             A- B = { x | xAxB }

Example 5: If A = {1, 2, 3} and B = {1, 2, 4, 5} ,  then A- B = {3} .
Example 6: If A = {1, 2, 3} and B = {4, 5} ,  then A- B = {1, 2, 3} .

Note that in general A- BB–A

·         Set Complimentary

The set of elements that are not in a set A is called the complement of A. It is written A′ (or sometimes A^C, or ).

Clearly, this is the set of elements that answer 'No' to the question Are you in A?.

For example, if U = N and A = {odd numbers}, then A′ = {even numbers}.

·         Characteristic of Set