Sets & Basic Operations

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Set Theory - 01

Sets & Basic Operations

Sets and Elements

A set is a well-defined collection of objects, each of which is called an element or member of the set.

Sets can either be defined by (i) listing out its members (such as the set of vowels a, e, i, o, u), or (ii) by stating out rules or properties (such as the set of the names of the capital cities of Indian states).

Notations

A set is normally denoted by a capital letter, such as A, X etc., while the elements might be denoted generically by lowercase letters like a, b etc. The elements of a set are enclosed within curly braces '{' and '}', and are separated from each other by commas.

Sets can be specified in two ways:

1. Tabular form:
The elements are listed explicitly. e.g., $A = { 2 , 4 , 6 , 8 }$ .
2. Set-builder notation or property method:
The set is defined by stating the properties which characterize the elements of the set. e.g.,
The above reads as "B is a set of x such that x is an odd integer and x > 0". x stands for any element of the set, the colon is read as "such that" and the comma as "and." In explicit form, the above set is $B = { 1 , 3 , 5 , 7 , 9 }$.
The symbolic way of writing the statement that an element "p is an element of set A" or, equivalently, "p belongs to set A," is $p ∈ A$
When specifying two elements, like p and q, we write $p , q ∈ A$
The statement "a does not belong to A" is written as $a ∉ A$
The set A of vowels can be defined as: $A = { x : x is a member of the english alphabet , x is a vowel }$
In the above set, note that $a , u ∈ A$ , while $b ∉ A$

Two sets A and B are equal if both have the same elements. This is denoted as A = B. If they are unequal, it is expressed as $A ≠ B$

Note that changing the arrangement of the elements of a set does not change the set. Also, repetition of elements of a set does not change the set.

Three sets are defined as follows: $A = { 2 , 4 , 6 , 8 }$, $B = { 6 , 2 , 8 , 4 }$ and $C = { 2 , 4 , 6 , 8 , 2 }$
A = B, since both contain the same elements, even though in different order.
A = C, since both contain the same elements, even though C has the element 2 repeated.

Universal Set and Empty Set

The universal set, or the universe, is a larger set assumed to contain all sets under consideration for a particular study. The universal set is denoted by U. Thus, for a study on car models, the universal set could consist of all the car models in the world ever made.

An empty set, or a null set, is a set with no elements. It is denoted by $∅$ or ${ }$. Evidently, there can be only one null set.

Subsets

If every element of a set A is also an element of set B, then A is called a subset of B. This is also expressed as: A is contained in B, or B contains A. This is expressed symbolically as $A ⊆ B$

Inversely, B is called a superset of A. Symbolically, this is expressed as $B ⊇ A$

The statement "A is not a subset of B" implies that at least one element of A does not belong to B. This is expressed as $A ⊈ B$ or $B⊉A$.

From the definition of subset, if A = B, A is still a subset of B, and vice versa.

Three sets are defined as follows:
$A = { 1 , 4 , 7 , 11 , 17 }$, $B = { 2 , 7 , 8 , 11 , 17 , 19 }$ and $C = { 7 , 17 }$
We can see that $C ⊆ A$ and $C ⊆ B$, as the elements 7 and 17 of C are also elements of A and B. However, $A ⊈ B$, since some elements of A are not elements of B.

Based on what we have learned till now, some properties of sets are

1. Every set A is a subset of the universal set U, since by definition, every element of a set A belongs to U.
2. The empty set $∅$ is a subset of any set A.
3. Every set A is a subset of itself. Thus, $A ⊆ A$
4. Two set are equal if and only if both are subsets of each other. In other words, A = B if and only if $A ⊆ B$ and $B ⊆ A$
5. If every element of a set A belongs to a set B, and every element of set B belongs to a set C, then clearly every element of set A belongs to set C. Thus, if $A ⊆ B$ and $B ⊆ C$, then $A ⊆ C$.
Proper Subset

If $A ⊆ B$ and $A ≠ B$, then A is a proper subset of B. This is expressed as $A ⊂ B$

Thus, all proper subsets are subsets, but not all subsets are proper subsets.

Disjoints sets

If two sets have no elements in common, they are said to be disjoint. Two sets which are disjoint can never have a superset-subset relationship, unless one is a null set.

Venn Diagrams

A venn diagram is a pictorial representation of sets, wherein they are shown as enclosed areas. Typically, the universal set U is represented by the area within a rectangle, and the other sets as circles placed within the rectangle.

Fig. represents a universal set and disjoint sets A and B, while Fig. represents a universal set with sets A and B having subset-superset relationship.

Set Operations

The three common operations on sets are the operations of union, intersection and difference.

Union

The union of two sets, A and B, denoted by $A ∪︀ B$, is the set of all elements which belong to either A or B; i.e.

$A ∪︀ B = { x : x ∈ A or x ∈ B }$

The venn diagram depicting the union relationship is shown in Fig. .

Intersection

The intersection of two sets, A and B, denoted by $A ∩︀ B$ , is the set of all elements which belong to both A and B; i.e.

$A ∩︀ B = { x : x ∈ A and x ∈ B }$

The venn diagram depicting the union relationship is shown in Fig. .

Three sets are defined as follows:
$A = { 1 , 4 , 7 , 10 }$, $B = { 2 , 4 , 6 , 8 , 10 , 12 }$ and $C = { 1 , 3 , 5 , 7 }$
 $A ∪︀ B = { 1 , 2 , 4 , 6 , 7 , 8 , 10 , 12 }$ $A ∩︀ B = { 4 , 10 }$ $B ∪︀ C = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 10 , 12 }$ $B ∩︀ C = ∅$ $A ∪︀ C = { 1 , 3 , 4 , 5 , 7 , 10 }$ $A ∩︀ C = { 1 , 7 }$
Let U denote the set of students in a class. Let set B and set G denote the collection of boys and girls respectively in the class.

We have,

$B ∪︀ G = U$, as each student in U is either in set B or set G.

$B ∩︀ G = ∅$, as there is no student belonging to both sets B and G.

Properties of unions and intersections

Some of the properties of the union and intersections of sets are as follows:

1. $( A ∩︀ B ) ⊆ A$ and $( A ∩︀ B ) ⊆ B$, since every element of $A ∩︀ B$ belongs to A, as well as B. This is also clear from the venn diagram of Fig. .
2. $A ⊆ ( A ∪︀ B )$ and $B ⊆ ( A ∪︀ B )$, since every element of A belongs to $A ∪︀ B$, and similarly every element of B belongs to $A ∪︀ B$. This is also clear from the venn diagram of Fig. .

Combining the above two properties leads to: $( A ∩︀ B ) ⊆ A ⊆ ( A ∪︀ B ) and ( A ∩︀ B ) ⊆ B ⊆ ( A ∪︀ B )$

Complement

The complement, or absolute complement, of a set A, denoted by $A c$ or $A ′$, is the set of all elements which belong to U (the universal set) but does not belong to A.

$A ′ = { x : x ∈ U , x ∉ A }$ (see Fig. )

Difference

The difference of two sets, A and B, denoted by A B, or A ~ B, is the set of all elements which belong to A but do not belong to B, i.e.

$A ∼ B = { x : x ∈ A , x ∉ B }$ (see Fig. )

Bibliography

Lipschutz, L, Schaum's Outline of Theory and Problems of Set Theory and Related Topics, 2nd edn, USA: McGraw Hill, 1998.
Nagel, R (ed), 'Set Theory,' U.X.L Encyclopedia of Science, 2nd edn, vol. 1-10, USA: The Gale Group, 2002.
Rosen, KH et al. (eds), Handbook of Discrete and Combinatorial Mathematics, Florida: CRC Press, 2000.