Union of Sets and its Complement

A union of 2 or more sets means that you are combining all the elements of all the sets into 1 new set. Typically, it is written as A ∪ B, for the case of 2 sets. For 3 sets, it's written as A ∪ B ∪ C, and so on for more sets.




The following Venn diagram represents this set operation with the shaded area, for the case of A ∪ B:



Here are a couple of properties for this operation. These are true only when all the operators in the problem are ∪:

The order of evaluation is not important, i.e. commutative.
A ∪ B = B ∪ A

Parenthesis does not really matter, i.e. associative.
(A ∪ B) ∪ C = A ∪ (B ∪ C)

Example
Given that A = {3, 4, 5, 8, 10}, B = {3, 5, 9, 12}, and C = {11, 13}.
(a) List all the elements of (i) set A or set B or both sets, (ii) set A or set B or set C or all the three sets.
(b) Find (i) A ∪ B, (ii) A ∪ B ∪ C
(c) Draw the Venn diagrams representing (i)A ∪ B, and (ii)A ∪ B ∪ C.

Solution
(a)
(i) 3, 4, 5, 8, 9, 10, 12
(ii) 3, 4, 5, 8, 9, 10, 11, 12, 13

(b)
(i) A ∪ B = {3, 4, 5, 8, 9, 10, 12}
(ii) A ∪ B ∪ C = {3, 4, 5, 8, 9, 10, 11, 12, 13}

(c)
(i)


(ii)



Complement of Union of Sets

The complement of A ∪ B is written as (A ∪ B)'. (A ∪ B)' simply refers to all the elements that are NOT in A ∪ B, but are still within the universal set, U. The simple Venn diagram below shows (A ∪ B)' as the shaded region.

The complement of A ∪ B is written as (A ∪ B)'. (A ∪ B)' simply refers to all the elements that are NOT in A ∪ B, but are still within the universal set, U. The simple Venn diagram below shows (A ∪ B)' as the shaded region.





Return from Union to Pre Algebra


Return Home to Algebra-by-Example.com

Find a Local Algebra Tutor Today
Loading

Find a Local Algebra Tutor Today