Set Complement and Set Difference

 Remember that we often work with a specific set of objects when solving problems or discussing issues. We called this set of objects a universal set or universe. For example, in the lead-in problem above, the universal set could be either the set of all U. S. dollars or the set of the $836 Sam originally had in the checking account.  

Complement of a Set: The complement of a set, denoted A', is the set of all elements in the given universal set U that are not in A. 

In set- builder notation,  A' = {x ∈ U : x ∉ A}.

The Venn diagram for the complement of set A is shown below where the shaded region represents A'. 
                                 

Example: For the lead-in example on the previous page, let the universal set U  be the $836 Sam originally has in the checking account and let A be the set of the $429 of the check. The complement of set A would be the set of the $407 remaining in the checking account. 

Example: Let U = {1, 2, 3, 4, 5, 6} and A = {1, 3, 5}. Then  A' = {2, 4, 6}.   

Example: U' = ∅   The complement of the universe is the empty set. 

Example: ∅'  = U  The complement of an empty set is the universal set.

Set Difference: The relative complement or set difference of sets A and B, denoted A – B, is the set of all elements in A that are not in B. 

In set-builder notation,  A – B = {x ∈ U : x ∈ A and  x ∉ B}= A ∩ A'.

The Venn diagram for the set difference of sets A and B is shown below where the shaded region represents A – B.
                                   

Example:  For the lead-in example on the previous page, let the universal set U be the set of all U.S. dollars, let set A be the set of $836 Sam originally has in the checking account, and let B be the set of the $429 of the check. Then the set difference of A and B would be the $407 remaining in the checking account. 

Example:  Let A = {a, b, c, d} and B = {b, d, e}.  Then A – B = {a, c} and B – A = {e}.

Example:  Let G = {t, a, n} and H = {n, a, t}. Then G – H = ∅.

How should we define the subtraction of whole numbers?  

        In the lead-in example on an earlier page of this section, the remaining balance was the difference between the cardinalities of the sets for the checking account and the check. This also works for the third example (above)where

n(G) – n(H) = 3 – 3 = 0 = n(∅).

But, with the second example (above) the difference between the cardinalities does not give the expected result, e.g.,

n(A) – n(B) = 4 – 3 = 1 ≠ 2 = n(A – B).

In this case, B is not a subset of A.