# Set Operations

Frequently we need to combine sets to produce new sets. For instance, say we have a set of animals that live under the water and another set of animals that live on the ground. We could make a new set of animals that can live both, under water and on the ground. Or a set of animals that cannot live in water.
There are a few ways of combining sets.

## 1. Intersections

Let's say that we have a set *A* = {1,2,3} and set *B* = {2,3,4}. **Intersection** of *A* and *B* is a new set - {2,3}. In other words, intersection is a set of objects that are present in multiple sets at the same time. In symbols, intersection is denoted by $\cap$.

$A \cap B \thickspace is \thickspace \{e| \space e \in A \land e \in B \}$
Example,

- {1,2,3} $\cap$ {2,3,4} = {2,3}

If two sets have no simultaneous objects, they are said to be **disjoint sets**.

Sometimes it is helpful to visualize sets to get the big picture. To do that we implement **Venn Diagrams**.

In the picture above, blue-purple area represents the intersection of *A* and *B*.

## 2. Union

Union is similar to intersection. With a small difference. When we want to represent the **union of ***A* and *B*, we take all the elements that appear in at least on of the sets. It does not include the duplicates. It is denoted by the symbol $\cup$.
In fancy symbols, we define it as the following:
$A \cup B \thickspace is \thickspace \{e| \space e \in A \lor e \in B \}$

For instance,

- {1,2,3,4} $\cup$ {3,5,6,7} = {1,2,3,4,5,6,7}

## 3. Symmetric Difference

**Symmetric difference** of *A* and *B* is a collection of objects that appear **only in ***A* or *B*. If an element is present in both sets, it cannot be a part of symmetric difference set. The special symbol to denote it is: $\oplus$.
$A \oplus B = \{ x| \space x \in A \oplus x \in B \}$

Example,

- {1,2,3} $\oplus$ {2,3,4,5} = {1,4,5}

Since 2 and 3 appear in both sets above, they are not in the symmetric difference set.

## 4. Complement

The **complement** of *B* **relative to** *A*, written as $A - B$ (not $B - A$), is the set of elements that are present in set *A* and not in set *B*.
$A - B = \{ x \mid x \in A \land x \notin B \}$
For example,

- {1,2,3,4} $-$ {1,2,5,6} = {3,4}

Elements 3 and 4 exist in set one but not in set two. Hence, they are called the complement of B with respect to A. Sometimes they are called the **difference** of A and B.