Difference of Sets

In this lesson we will learn about another binary operation called Difference.

The Difference of $A$ from $B$ is denoted by $A-B$ which is the set of all elements in $A$ that are not in $B$.

Example

This is simply taking out any elements that are in $B$ out of A.

Properties of difference

The following properties should be intuitive:

• The difference of a set from a disjoint set is the same set. $A-B = A$ if $A \cap B = \phi$
• The difference of a subset from its set is the null set. $A-B = \phi$ if $A \subseteq B$
• The difference of a set from the empty set is the same set. $A-\phi = A$.
• The difference of a set from the universal set is the empty set. $A-U = \phi$
• Generally, difference is not commutative. $A-B \ne B-A$
• The difference of a Universal set from a set is the complement of that set. $U-A = A^c$

Non-Commutativity of Difference

Let's look at some examples to confirm that difference is non commutative.

The first example is the difference of a set and the universal set. $A-U$ is the set of all elements of $A$ that are not in $U$ which is the empty set. But on the other side, $U-A$ is the set of all elements of the universal set which are not in $A$ which is the definition of the complement of a set.

Other explicit examples are;

Symbolic Representation of Difference

To do some practice of set builder notation, we can write the difference like this.

Venn Diagram Visualization