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# 10.1: Sets and Set Notation

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A set is a collection of things called elements. For example $$\left\{ 1,2,3,8\right\}$$ would be a set consisting of the elements 1,2,3, and 8. To indicate that $$3$$ is an element of $$\left\{ 1,2,3,8\right\} ,$$ it is customary to write $$3\in \left\{ 1,2,3,8\right\}.$$ We can also indicate when an element is not in a set, by writing $$9\notin \left\{ 1,2,3,8\right\}$$ which says that $$9$$ is not an element of $$\left\{ 1,2,3,8\right\}.$$ Sometimes a rule specifies a set. For example you could specify a set as all integers larger than $$2.$$ This would be written as

$S=\left\{ x\in \mathbb{Z}:x>2\right\} .\nonumber$

This notation says: $$S$$ is the set of all integers, $$x,$$ such that $$x>2.$$

Suppose $$A$$ and $$B$$ are sets with the property that every element of $$A$$ is an element of $$B$$. Then we say that $$A$$ is a subset of $$B.$$ For example, $$\left\{ 1,2,3,8\right\}$$ is a subset of $$\left\{ 1,2,3,4,5,8\right\}.$$ In symbols, we write

$\left\{ 1,2,3,8\right\} \subseteq \left\{ 1,2,3,4,5,8\right\} .\nonumber$

It is sometimes said that “$$A$$ is contained in $$B$$" or even “$$B$$ contains $$A$$". The same statement about the two sets may also be written as

$\left\{ 1,2,3,4,5,8\right\} \supseteq \left\{ 1,2,3,8\right\}. \nonumber$

We can also talk about the union of two sets, which we write as $$A \cup B$$. This is the set consisting of everything which is an element of at least one of the sets, $$A$$ or $$B$$. As an example of the union of two sets, consider

$\left\{ 1,2,3,8\right\} \cup \left\{ 3,4,7,8\right\} =\left\{ 1,2,3,4,7,8\right\}.\nonumber$

This set is made up of the numbers which are in at least one of the two sets.

In general

$A\cup B = \left\{ x:x\in A \text{ or }x\in B\right\}\nonumber$

Notice that an element which is in both $$A$$ and $$B$$ is also in the union, as well as elements which are in only one of $$A$$ or $$B$$.

Another important set is the intersection of two sets $$A$$ and $$B$$, written $$A \cap B$$. This set consists of everything which is in both of the sets. Thus $$\left\{ 1,2,3,8\right\} \cap \left\{ 3,4,7,8\right\} =\left\{ 3,8\right\}$$ because $$3$$ and $$8$$ are those elements the two sets have in common. In general, $A\cap B = \left\{ x:x\in A\text{ and }x\in B\right\}\nonumber$

If $$A$$ and $$B$$ are two sets, $$A\setminus B$$ denotes the set of things which are in $$A$$ but not in $$B.$$ Thus $A\setminus B = \left\{ x\in A:x\notin B\right\}\nonumber$ For example, if $$A = \left\{1,2,3,8 \right\}$$ and $$B = \left\{ 3,4,7,8 \right\}$$, then $$A \setminus B = \left\{ 1,2,3,8\right\} \setminus \left\{ 3,4,7,8 \right\} =\left\{1,2 \right\}$$.

A special set which is very important in mathematics is the empty set denoted by $$\emptyset$$, which is defined as the set which has no elements in it. It follows that the empty set is a subset of every set. This is true because if it were not so, there would have to exist a set $$A,$$ such that $$\emptyset$$ has something in it which is not in $$A.$$ However, $$\emptyset$$ has nothing in it and so it must be that $$\emptyset \subseteq A.$$

We can also use brackets to denote sets which are intervals of numbers. Let $$a$$ and $$b$$ be real numbers. Then

• $$\left[ a,b\right] = \{x \in \mathbb{R}: a\leq x\leq b \}$$
• $$\left[a,b \right) = \{x \in \mathbb{R}: a\leq x<b \}$$
• $$\left( a,b\right) = \{x \in \mathbb{R}: a<x<b \}$$
• $$(a,b] = \{ x \in \mathbb{R}: a<x\leq b \}$$
• $$[a,\infty ) = \{x \in \mathbb{R}: x\geq a \}$$
• $$(-\infty ,a] = \{x \in \mathbb{R}: x \leq a \}$$

These sorts of sets of real numbers are called intervals. The two points $$a$$ and $$b$$ are called endpoints, or bounds, of the interval. In particular, $$a$$ is the lower bound while $$b$$ is the upper bound of the above intervals, where applicable. Other intervals such as $$\left( -\infty ,b\right)$$ are defined by analogy to what was just explained. In general, the curved parenthesis, $$($$, indicates the end point is not included in the interval, while the square parenthesis, $$[$$, indicates this end point is included. The reason that there will always be a curved parenthesis next to $$\infty$$ or $$-\infty$$ is that these are not real numbers and cannot be included in the interval in the way a real number can.

To illustrate the use of this notation relative to intervals consider three examples of inequalities. Their solutions will be written in the interval notation just described.

Example $$\PageIndex{1}$$: Solving an Inequality

Solve the inequality $$2x+4\leq x-8$$.

Solution

We need to find $$x$$ such that $$2x+4\leq x-8$$. Solving for $$x$$, we see that $$x\leq -12$$ is the answer. This is written in terms of an interval as $$(-\infty ,-12].$$

Consider the following example.

Example $$\PageIndex{2}$$: Solving an Inequality

Solve the inequality $$\left( x+1\right) \left( 2x-3\right) \geq0.$$

Solution

We need to find $$x$$ such that $$\left( x+1\right) \left( 2x-3\right) \geq0.$$ The solution is given by $$x\leq -1$$ or $$x\geq \frac{3}{2}$$. Therefore, $$x$$ which fit into either of these intervals gives a solution. In terms of set notation this is denoted by $$(-\infty ,-1]\cup [ \frac{3}{2},\infty ).$$

Consider one last example.

Example $$\PageIndex{3}$$: Solving an Inequality

Solve the inequality $$x \left( x+2\right) \geq-4$$.

Solution

This inequality is true for any value of $$x$$ where $$x$$ is a real number. We can write the solution as $$\mathbb{R}$$ or $$\left( -\infty ,\infty \right) .$$

In the next section, we examine another important mathematical concept.