Difference

Set difference

If we have two sets, $A$ and $B$, the difference, $A−B$, is the set containing all the elements that are members of $A$ and not members of $B$. Some authors also denote this difference as $A \setminus B$. We can represent it mathematically as follows:

Because the difference $A−B$ contains all the elements of $A$ except those that are common with $B$, we can infer the following:

Examples

Let’s explore a few examples.

For the first example, let’s consider the following sets:

To find the members of $A$ that are not in $M$, we look at their set difference as follows:

To find the members of $M$ that are not members of $A$, we derive the difference as follows:

Now, let’s look at another example. In the English alphabet, each letter is a vowel or a consonant, but not both. We can represent this information using set-theoretic notation, as shown below:

The following set of consonants contains the letters from the alphabet that are not vowels:

Similarly, we can tell that $V = E\setminus C$.

Properties of set differences