# Sets Overview

A set is essentially a collection of items, where each item is considered an **element**. Elements are listed inside of brackets, and as a collection, they make up a **set**. Sets are usually named using a capital letter. For example, \(A=\{1,2,3\}\) states that set \(A\) consists of the elements \(1,2,\) and \(3\). You know that a number is considered an element of a set if it is listed inside of the brackets. For example, \(5\) is an element of the set \(T=\{5,6,7\}\), but \(4\) is not an element of this set.

Sometimes the elements of a set will be words or items, instead of numbers. For example, we can list the elements of the set “things on the kitchen table” as K={plates, cups, silverware, table cloth, water bottle, candle}.

Sets are often used in algebra when writing the solutions for **inequalities**. For example, the inequality \(x \lt 4\) means that all values less than \(4\) are solutions for \(x\). You will often need to phrase your answer in what is referred to as “set notation”. So instead of saying “all values less than \(4\) are solutions for \(x\)”, we use a consistent collection of symbols to describe the set of solutions. The solutions for \(x \lt 4\) would be expressed as \(\{x \mid x\text{ is a real number and} x \lt 4\}\) using set notation. Solution sets often consist of many values. Using set notation saves space and condenses the solution set into a more compact format.

There are a few symbols to be familiar with when studying set notation. Here are the symbols you should be most familiar with:

\(\{\text{ }\}\) “is a set”

\( \epsilon \) “is an element of”

Sometimes sets can be **subsets** of other sets. For example, the set \(A=\{3,4,5\}\) is a subset of \(B=\{2,3,4,5,6,7\}\) because all of the elements of set \(A\) are also in set \(B\).

\(B=\{2,3,4,5,6,7\}\)

The symbol for “subsets” is \(\subset\). This is read as “is a subset of”.

Sometimes sets will contain elements that are short and concise like \(A=\{1,2,3,4,5\}\). However, some sets will contain elements that are never ending. These sets are referred to as **infinite sets**.

Using set notation for an infinite set is very similar to the notation for a finite set. The only difference is the use of the ellipsis (…).

For example, \(C=\{2,3,4\}\) represents a finite set, but \(D=\{4,5,6…\}\) represents an infinite set. One way to think about finite and infinite sets is to think of finite sets as the factors of a number, and infinite sets as the multiples of a number. Factors are limited, but multiples can continue on infinitely.

**Sets Sample Questions**

Here are a few sample questions going over sets.

**Question #1:**

Which of the following is an infinite set?

**Answer:**

An infinite set consists of so many elements that it is simply “uncountable”. The use of the ellipses (…) indicates an infinite set.

**Question #2:**

Which of the following is a subset of \(Y\)?

\(Y=\{2,4,6,8,10,12\}\)

**Answer:**

\(R\) is a subset of \(Y\) because all of the elements in set \(R\) are also in set \(Y\).

\(Y=\{2,4,\mathbf{6},\mathbf{8},\mathbf{10},12\}\)

**Question #3:**

Which of the following is true about set \(R\)?

\(R=\{-3,-2,-1,0,1,2,3\}\)

\(R\) is an infinite set.

All of the above are true.

**Answer:**

\(-2\) is an element of the set. In set notation this is expressed as \(-2 \epsilon R\). \(-4\) is not listed in the set, and the set is not infinite.

**Question #4:**

What is the meaning of \(P=\{2,4,6,8,10\}\), \(2 \epsilon P\)?

Set \(P\) is infinite

\(2\) is not an element of \(P\)

\(P\) is an element of \(2\)

\(2\) is an element of \(P\)

**Answer:**

The symbol \(\epsilon\) means that the value is an element of the set. In this case, \(2\) is an element of set \(P\), so it is written as \(2 \epsilon P\).