Sets and Venn Diagrams (Economics)

Set Theory

Set theory is the branch of mathematics which studies sets.

A set is a collection of elements (or objects) and can be considered as an element itself. We often denote a set using a capital letter. Sets are used to group and describe elements which share a similar property. If an element $x$ is included in a set we say that $x$ is a member of that set. Sets are called disjoint if they have none of the same elements.

The sample space (or universal set) is a set which includes all elements that a set may contain. It is typically denoted using the letter $S$. If we have only one set $A$, then the sample space is the same as that set: $S=A$.

The empty (or null) set $\varnothing$ contains no elements at all.

Defining a Set

There are $2$ main ways in which we can define a set. These are known as the listing (or roster) method and the rule method.

Listing Method

This method involves writing the members of a set as a list, separated by commas and enclosed within curly braces. For example, the four seasons are a set and could be written as {Summer, Autumn, Spring, Winter}.

Note: The order of the elements in the list doesn’t matter. For example, we could have written the set of seasons as {Spring, Autumn, Summer, Winter} or {Winter, Autumn, Spring, Summer}.

Rule Method

This method involves specifying a rule or condition which can be used to decide whether an object can belong to the set. This rule is written inside a pair of curly braces and can be written either as a statement or expressed symbolically or written using a combination of statements and symbols.

The rule method is often preferred when defining larger sets where it would be difficult or time consuming to list all of the elements in a set.

For example, a set $A$ whose elements included all the positive integers could be defined symbolically as: \[A=\{x|x\gt0\}\] or in statement form as: \[A=\{x|x \text{ is a positive number}\}\] The set $B$, whose elements include all the natural numbers from $1$ to $25$ inclusive could be defined symbolically as: \[B=\{x|x \in \mathbb{N},1\leq x \leq 25\}\] or using a statement and symbols as: \[B=\{x|x \text{ is a natural number}, 1\leq x \leq 25\}\]

Types of Sets

Finite Sets

Finite sets are sets which contain a limited and countable number of elements. The sets $A=\{1,2,6\}$ and $B=\{x|1\leq x \leq 10\}$ are both examples of finite sets.

Infinite Sets

Infinite sets contain an unlimited (or infinite) and uncountable number of elements. For example, the sets $C=\{2, 4, 6, ...\}$ and $D=\{x|x\leq 0\}$ are both infinite sets.

Venn diagrams

Venn diagrams are a useful way of visualizing relationships between sets.

When we draw Venn diagrams, sets are represented by circles. The inside of a circle represents all the elements that are members of that set and the outside of the circle represents all elements which are not members of that set.

The areas where circles overlap represents the set of elements which are in all of the sets corresponding to the overlapping circles. We call this area the intersection of the (overlapping) sets.

The sample space is represented by a rectangle which is drawn around all of the circles and we write the set name of the sample space (typically the letter $S$) in the left-hand corner. If we want to partition the sample set into multiple parts, we draw lines to divide the rectangle into the required number of parts.

Set Notation

Suppose that our sample space is the set of natural numbers $\mathbb{N}=\{1, 2, 3, 4, ...\}$. Let $A$ be the set containing the numbers $1$ to $5$ (inclusive), $B$ be the set containing the numbers $1$ to $3$ (inclusive) and $C$ be the set containing the numbers $1$, $2$ and $6$; that is we have $A=\{1, 2, 3, 4, 5\}$, $B=\{1, 2, 3\}$ and $C=\{1, 2, 6\}$. We can represent the following relationships as follows:

Symbol	Pronounciation	Interpretation	Set	Venn Diagram
$A^c$	$A$ complement	Every element which is not a member of $A$	$\{6, 7, 8,...\}$
$2 \in A$	$2$ is an element of $A$	The number $2$ is a member of the set $A$
$-4 \notin C$	$-4$ is not an element of $C$	The number $-4$ is not a member of the set $C$
$A \supset B$	$A$ contains $B$	Every element of $B$ is also an element of $A$
$B \subset A$	$B$ is a subset of $A$	Every element of $B$ is also an element of $A$
$A \cup C$	$A$ union $C$	Every element which is a member of $A$ OR $C$	$\{1, 2, 3, 4, 5, 6\}$
$A \cap C$	$A$ intersect $C$	Every element which is a member of both $A$ AND $C$	$\{1, 2\}$

Laws of Operations on Sets

The laws of operations on sets are very similar to the fundamental laws of algebra, with the operations addition and multiplication replaced by union and intersection.

Necessity and Sufficiency Applied to Sets

We can use the concepts of necessary and sufficient conditions to describe relationships between sets.

For example, the set of natural numbers $\mathbb{N}$ is a subset of the set of real numbers $\mathbb{R}$, since every natural number is a real number. In other words, being in the set $\mathbb{N}$ is a sufficient but not a necessary condition for being in the set $\mathbb{R}$. We can write this as: \[\mathbb{N} \Rightarrow \mathbb{R}.\]

Another way to think about the relationship between the sets $\mathbb{R}$ and $\mathbb{N}$ is that being in the set $\mathbb{R}$ is a necessary but not a sufficient condition for being in the set $\mathbb{N}$. For example, $3.5$ is a real number but not a natural number. We thus have \[\mathbb{R} \Leftarrow \mathbb{N}.\]

Note: In set notation we would write the relationship between these two sets as $\mathbb{N} \subset \mathbb{R}$ or $\mathbb{R} \supset \mathbb{N}$.

External Resources

• Sets and venn diagrams by Maths is Fun
• Questions on venn diagrams by Regents Exam Prep Center
• Logic Problems and Venn Diagrams by The Centre for Innovation in Mathematics Teaching