Concepts and Notations
In mathematics, the words “collection”, “class”, or “set” are interchangeable. However, the word “set” is most commonly used.
- A “set” is a gathering or collection of distinct objects, which are called elements (or members) of the set.
- The number of elements can be finite, infinite, or even none.
- Sets are usually designated by capital letters $A,B,C,\ldots,X,Y,Z$ and elements by lower-case letters $a,b,c,\cdots,x,y,z$.
- A simple way to specify a set is to list its elements (when it is possible). To this end, we put the elements inside a pair of braces (curly brackets) $\{\quad\}$ . For example $\{1,-5,0\}$ is a set whose elements are 1, $-5$ and 0.
- Sometimes when writing all the elements of a set is not feasible, we may skip some of the elements and replace them by ellipsis. For example, to show the set of positive integers from 1 to 50, we can write $\{1,2,3,…,50\}$.
- The ellipsis is used only when the elements follow a pattern that can be clearly understood from the first few elements.
- If there is no elements after the ellipsis, it is assumed that the pattern continues forever. For example the set written as $\{1,2,3,4,5,\ldots\}$ contains all the positive integers. To show that the elements of a set go on forever in both directions, we may use the ellipsis at both ends. For example, the set of all integers (positive, negative and zero) may be displayed as $\{…,-4,-3,-2,-1,0,1,2,3,4,…\}$.
- Changing the order of the elements, or repeating the elements does NOT change the set. For example, the set $\{a,a,b,b,b,c,c\}$ is the same as $\{b,a,c\}$.
- When the object $x$ is an element of the set $A$, we write \[x\in A.\] Alternatively we can say “$x$ is in $A$”, “$x$ belongs to $A$”, or “$A$ contains $x$.” When $x$ is not an element of $A$, we write \[x\not\in A.\]
For example, let $A=\{1,2,3\}$. Here $A$ is a set whose elements are 1, 2, and 3. In this example, $1\in A$ but $5\not\in A$.
Subsets
We say a set $B$ is a subset of a set $A$, and we write
\[B\subset A\]
if every element of $B$ also belongs to $A$. In other words, for every $x$
\[x\in B\quad\quad\Rightarrow\quad\quad x\in A.\]
We can also say “$A$ contains $B$” or “$B$ is contained in $A$.”
For example, if $A=\{-1,3,8,11,9\}$, $B=\{3,9,11\}$, and $C=\{3,8,2\}$ then
\[B\subset A\]
because every element of $B$ is also an element of $A$, but $C$ is not a subset of $A$
\[C\not\subset A\]
because $2\in C$ but $2\not\in A$.
Equality
We say two sets $A$ and $B$ are “equal” or “identical”, and we write $A=B$, if they consist of exactly the same elements.
- If two sets $A$ and $B$ are equal then $A\subset B$ and $B\subset A$. If $A\subset B$ and $B\subset A$, then $A=B$. That is,
\[A=B\quad\Leftrightarrow\quad A\subset B\text{ and }B\subset A.\]
Universal Set
A set that contains all the elements that we want to consider for now is called the “universal set”, usually denoted by $U$ (or in some books by $S$). For example, we might say that the universal set is the set of all real numbers or the set of all integers.
The universal set is also called the “domain of discourse” or the “universe of discourse“. The universal set may vary from one application to another.
In elementary calculus, the universal set is assumed to be the set of all real numbers unless otherwise stated.
Set Builder Notation
When listing of all elements of a set is not possible, we can specify the set by describing a property common to the elements in the set and only to those elements. For example:
$A=$ the set of all numbers which are positive odd integers
or
$B=$ the set of all real numbers $x$ for which $-1<x<2$.
When we want to describe a set in this way, we can use set-builder notation. The general form of set-builder notation is
\[\{x\in U|\ x\text{ has property }P\}\]
that designates the set of all elements $x$ in $U$ for which the property $P$ is satisfied. The vertical bar which can also be written as a colon “:”, is a separator that is used in place of “such that.”
As just mentioned, in calculus the universal set when not specified is assumed to be the set of real numbers, so we may omit the reference to $U$ and simply write
\[\{x|\ x\text{ has property }P\} \]
For example,
$\{x|\ 1<x<2\}$= set of all real numbers $x$ which are greater than 1 but less than 2.
$\{x|\ x^{2}-3x+2=0\}$= set of all numbers $x$ for which $x^{2}-3x+2=0$.
In the above examples, $x$ is a “variable.” A variable is a symbol that represents any element of a given set. You may read more about variables in here and here.
Because it does not matter which symbol we use to represent the elements of a set, the letter $x$ is a dummy and can be replaced by any other symbol. Thus we may write
\[\{x|\ 1<x<2\}=\{s|\ 1<s<2\}=\{p|\ 1<p<2\}.\]
Empty Set or Void Set
A set is empty if it has no elements. The empty set or void set is denoted by $\varnothing$
\[\varnothing=\{\}.\]
- Note that a box that contains only an empty box is not empty. Similarly $\{\{ \}\}$ is not an empty set, because it has one element: $\{ \}$.
\[\{\}\in\{\{\}\}.\] - For every set $A$:
\[\varnothing\subset A\]
Venn Diagrams
Venn diagrams, introduced by the English mathematician John Venn, are useful for visualizing sets and relations between them. In a Venn diagram, we represent a set as a region, often a disk, in the plane and its elements as points. The universal set, conventionally represented by a rectangle, is the outmost shape. If two regions overlap, it means that the two corresponding sets have some elements in common. The following Venn diagrams illustrate the relations we have discussed so far.
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| (a) $x\in A$ | (b) $x\not\in A$ |
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| (c) $B\subset A$ | (d) $B\not\subset A$ |
Union
If $A$ and $B$ are two sets, the union of $A$ and $B$, written as $A\cup B$ is the set that contains all elements of $A$ and $B$; that is, a set whose elements belong to $A$ or $B$ or both
\[ A\cup B=\{x|\ x\in A\quad\text{or}\quad x\in B\}. \]
In in the following figure, the shaded portion represents $A\cup B$.
Intersection
The intersection of two sets, $A$ and $B$, is the set whose elements belong to both $A$ and to $B$. The intersection of $A$ and $B$ is denoted by $A\cap B$:
\[A\cap B=\{x|\ x\in A\quad\text{and}\quad x\in B\}.\]
In the following figure, the shaded portion represents $A\cap B$.
Set Difference
Let $A$ and $B$ be two sets. The difference $A-B$ is the set of all elements $x$ in $A$ that are not in $B$:
\[A-B=\{x|\ x\in A\quad\text{and}\quad x\notin B\}. \]
The shaded portion in the Venn diagram in the following figure (a) represents the sets $A-B$ and that in (b) represents $B-A$.
- Some books use a back slash $\backslash$ and write $A\backslash B$ to emphasize that the operation of set difference is different from the ordinary idea of subtraction.
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| (a) $A-B$ | (b) $B-A$ |