1 Sets

A set is any collection of objects. Here, the word ‘object’ is used in its most general sense: an object may be a diode, an aircraft, a number, or a letter, for example.

A set is often described by listing the collection of objects - these are the members or elements of the set. We usually write this list of elements in curly brackets, and denote the full set by a capital letter. For example,

$C=\left\{\text{the resistors produced in a factory on a particular day}\right\}$

$D=\left\{\text{on, off}\right\}$

$E=\left\{0,1,2,3,4,5,6,7,8,9\right\}$

The elements of set $C$ , above, are the resistors produced in a factory on a particular day. These could be individually labeled and listed individually but as the number is large it is not practical or sensible to do this. Set $D$ lists the two possible states of a simple switch, and the elements of set $E$ are the digits used in the decimal system.

Sometimes we can describe a set in words. For example,

‘ $A$ is the set all odd numbers’.

Clearly all the elements of this set $A$ cannot be listed.

Similarly,

‘ $B$ is the set of binary digits’ i.e. $B=\left\{0,1\right\}$ .

$B$ has only two elements.

A set with a finite number of elements is called a finite set . $B,C,D$ and $E$ are finite sets. The set $A$ has an infinite number of elements and so is not a finite set. It is called an infinite set .

Two sets are equal if they contain exactly the same elements. For example, the sets $\left\{9,10,14\right\}$ and $\left\{10,14,9\right\}$ are equal since the order in which elements are written is unimportant. Note also that repeated elements are ignored. The set $\left\{2,3,3,3,5,5\right\}$ is equal to the set $\left\{2,3,5\right\}$ .

1.1 Subsets

Sometimes one set is contained completely within another set. For example if $X=\left\{2,3,4,5,6\right\}\text{ and }Y=\left\{2,3,6\right\}$ then all the elements of $Y$ are also elements of $X$ . We say that $Y$ is a subset of $X$ and write $Y\subseteq X$ .

Example 1

Given $A=\left\{0,1,2,3\right\}$ , $B=\left\{0,1,2,3,4,5,6\right\}$ and $C=\left\{0,1\right\}$ , state which sets are subsets of other sets.

Solution

$A$ is a subset of $B$ , that is $A\subseteq B$

$C$ is a subset of $B$ , that is $C\subseteq B$

$C$ is a subset of $A$ , that is $C\subseteq A$ .

Task!

A factory produces cars over a five day period; Monday to Friday. Consider the following sets,

1. $A$ = {cars produced from Monday to Friday}
2. $B$ = {cars produced from Monday to Thursday}
3. $C$ = {cars produced on Friday}
4. $D$ = {cars produced on Wednesday}
5. $E$ = {cars produced on Wednesday or Thursday}

State which sets are subsets of other sets.

Answer

1.2 The symbol $\in$

To show that an element belongs to a particular set we use the symbol $\in$ . This symbol means is a member of or ‘belongs to’. The symbol $\notin$ means is not a member of or ‘does not belong to’.

For example if $X=\left\{\text{all even numbers}\right\}$ then we may write $4\in X$ , $6\in X$ , $7\notin X$ and $11\notin X$ .

1.3 The empty set and the universal set

Sometimes a set will contain no elements. For example, suppose we define the set $K$ by

$K=\left\{\text{all odd numbers which are divisible by 4}\right\}$

Since there are no odd numbers which are divisible by 4, then $K$ has no elements. The set with no elements is called the empty set , and it is denoted by $\varnothing$ .

On the other hand, the set containing all the objects of interest in a particular situation is called the universal set , denoted by $S$ . The precise universal set will depend upon the context. If, for example, we are concerned only with whole numbers then $S=\left\{\cdots -5,-4,-3,-2,-1,0,1,2,3,4,5,\dots \right\}$ . If we are concerned only with the decimal digits then $S=\left\{0,1,2,3,4,5,6,7,8,9\right\}$ .

1.4 The complement of a set

Given a set $A$ and a universal set $S$ we can define a new set, called the complement of $A$ and denoted by $A^{\prime }$ . The complement of $A$ contains all the elements of the universal set that are not in $A$ .

Example 2

Given $A=\left\{2,3,7\right\}$ , $B=\left\{0,1,2,3,4\right\}$ and $S=\left\{0,1,2,3,4,5,6,7,8,9\right\}$ state

1. $A^{\prime }$
2. $B^{\prime }$

Solution

1. The elements of $A^{\prime }$ are those which belong to $S$ but not to $A$ .

   $A^{\prime }=\left\{0,1,4,5,6,8,9\right\}$

2. $B^{\prime }=\left\{5,6,7,8,9\right\}$

Sometimes a set is described in a mathematical way. Suppose the set $Q$ contains all numbers which are divisible by 4 and 7. We can write

$Q=\left\{x:x\text{ is divisible by 4 and }x\text{ is divisible by 7}\right\}$

The symbol : stands for ‘such that ’. We read the above as ‘ $Q$ is the set comprising all elements $x$ , such that $x$ is divisible by 4 and by 7’.