Set operations and Venn Diagrams

When you work with abstract concepts, visualizing them can help a lot. In this topic, we'll present a well-known way to visualize sets.

Euler diagrams

To show relations between sets, one can use Euler diagrams (sometimes also called Euler-Venn diagrams). You can start drawing an Euler diagram by drawing a rectangle that represents the universal set $U$, which contains all the objects under consideration. Note that the universal set, or universe, varies depending on which objects are of interest. Inside this rectangle, you can use circles (or, more rarely, other geometrical figures) to represent sets. You can also mark points on the diagram, representing the particular elements of the set.

For example, the universe $U$ may be the set of all integers or the set of all fruits and berries. Let's proceed with the second example. Suppose that$A = \{\text{Lemon, Banana, Mango, Apple, Pineapple}\}$ and $B = \{\text{Cranberry, Gooseberry, Raspberry, Blueberry, Currant}\},$To represent these sets graphically, you can use the following Euler diagram:

If your universe is the set of school subjects and the sets are $P = \{\text{History, English}\}$, $Q = \{\text{English, Russian}\}$, $R = \{\text{Biology, English}\}$, you can draw them on a diagram:

Set intersection

Suppose you have two sets: $A = \{1, 2, 3\}$ and $B = \{2, 3, 5\}$. You can denote the intersection operation by $\cap$. The set $A \cap B$, the intersection of $A$ and $B$, is the set that contains all elements that belong to both $A$ and $B$. Thus, you can write $A \cap B = \{2, 3\}$. Again, you can use an Euler diagram to show this graphically:

These circles overlap. One circle would be the set $A$, and the other circle would be the set $B$.

Since the intersection is a set that contains only the elements that are in both sets, it is just the area where two circles overlap. It means that the set $A \cap B$ is the overlapping area.

Thus, $A \cap B = \{x \ | \ x \in A \ \text{and} \ x \in B\}$

If you have two circles that don't overlap at all, then their intersection is an empty set.

For example, if $A = \{\text{Python, C++}\}$, $B = \{\text{Scala, R}\}$, then $A \cap B = \emptyset$. Again, let's look at it on an Euler diagram:

As you can see, there is no overlap between $A$ and $B$.

Set union

Suppose you have a set $A = \{0, 1, 2\}$ and a set $B = \{-1, -2, -3, 1, 2\}$. The symbol $\cup$ denotes the union of two sets. The set $A \cup B$, the union of $A$ and $B$, is the set that contains all elements belonging to either the set $A$, or the set $B$, or both. Thus, $A \cup B = \{ 0, 1, 2, -1, -2, -3\}$. To imagine it visually, look at the Euler diagram:

Everything in the set $A$, everything in the set $B$, and the overlapping area belong to the set $A \cup B$.

Thus, $A \cup B = \{x \ | \ x \in A \ \text{or} \ x \in B\}$

If two sets don't overlap, for instance, $A = \{\text{Sun, Moon}\}$ and $B = \{\text{Earth, Mars}\}$, then $A \cup B = \{\text{Sun, Moon, Earth, Mars}\}$. In this case, you would have the following Euler diagram:

Set complement

Suppose you have a universal set $U = \{1, 2, 3, \ldots, 10\}$ and a set $A = \{1, 3, 5, 7, 9\}$. The complement of $A$ (which is denoted by $A'$ or $\overline{A}$) is a set of all elements in the $U$ that are not in $A$. In your case, $\overline{A} = \{2, 4, 6, 8, 10\}$. The following Euler diagram shows this situation graphically:

$\overline{A}$ is everything that is in the $U$, but outside the circle.

Hence, $\overline{A} = \{x \ | \ x \in U \ \text{and} \ x \not \in A\}$

Sets difference, or relative complement

You have a set $A = \{1, 2, 3, 4, 5\}$ and a set $B = \{4, 5, 6, 7\}$. If you choose all elements that are in the set $A$ but are not in the set $B$, you will get a set $A\backslash B = \{1, 2, 3\}$: it is the difference, or the relative complements, of $B$ in $A$. Look at the following Euler diagram that show it graphically:

Thus, $A \backslash B = \{x \ | \ x \in A \ \text{and} \ x \not \in B\}$

We can see that $A \setminus B \neq B \setminus A$.

Symmetric difference

The symmetric difference of sets $A$ and $B$ is the set of all elements belonging to one and only one of these sets. For example, for$A = \{1, 2, 3, 4, 5\}$ and $B = \{4, 5, 6, 7\}$ the symmetric difference is

$A \triangle B = \{1, 2, 3, 6, 7\}$The elements $1, 2, 3$ belong only to $A$ and the elements $6, 7$ only to $B$. You can see that

$A \triangle B = (A \setminus B) \cup (B \setminus A)$

Another equivalent formula for set difference is

$A \triangle B = (A \cup B) \setminus (A \cap B)$showing that you can take the union of the sets and throw away the elements belonging to both sets (that is, the elements from their intersection).

Conclusion

Euler diagrams allow you to organize information visually, and with them, you can clearly see the relations between two (or more) sets of items. People use them across many other disciplines. In this topic, you have learned about operations with sets (intersection, union, set complement, and set difference) and looked at how Euler diagrams can represent them. The set theory underlies many concepts, such as logics, hence it's crucial to understand operations with sets and to be able to visualize them.