Sets

[mashshare]

A set is nothing but a collection of well-defined objects. The objects present in sets are called “elements” or “members”.

Properties of Set

(i) All the elements in a set need to have a special property.

(ii) These elements also need to be different from each other and should not repeat.

(iii) They need to be well defined.

When we write the name of any set, it’s expressed in capital letters of English Alphabets such as:

A, B, C, D…………………… X, Y, Z

However, the elements and members in a set are written in small letters of English Alphabets. They are also written with the help of a number system within a brace (medium bracket) with using commas { }

Example: A = {a, b, c, d} or B = {1, 2, 3, 4}

Standard Symbols of Some Special Sets

N: the set of all natural numbers
Z: the set of all integers
Q: the set of all rational numbers
R: the set of real numbers
Z+: the set of positive integers
Q+: the set of positive rational numbers
R+: the set of positive real numbers.

Symbols and Their Meanings 

(i) “” (epsilon): “belong to” 

If a is an element of a set A, it is expressed that “a belongs to A”

a ∈ A

(ii) ” ”: “not belong to” 

In a situation when an element does not belong to any given set, then we apply symbol ∉: “not belong to”

But we say that “a not belong to A” if a is not an element of set A,

When it comes to the symbolic form, it’s written as: a ∉ A

(iii) ” ”: emplies 

In common language “” means “emply toward ….. Also ……”

(iv) ” = “: Equal to

A = B, it means set A is equal to set B.

(v) ” ≠”: Not equal to 

A ≠ B, It means set A not equal to set B.

(vi) “” is a symbol that is used for two-way implications, and is usually read as “if and only if “.

(vii) “: “: colon… in this set, the colon stands for “such that”

Representation of Sets 

Every set is always introduced by its elements. So, when it comes to expressing a set, we have to express its elements before.

Here are the methods used for expressing the elements of a set.

  • Roster or tabular form

When it comes to the roster form, the elements of a set are given, being separated by commas and enclosed within braces { }. For instance,

(i) First five natural numbers:

A = {1, 2, 3, 4, 5}.

(ii) First four letter of English Alphabets

B = {a, b, c, d}.

(iii) The set of all natural numbers that divide 42

C = {1, 2, 3, 6, 7, 14, 21, 42}.

(iv) The set of all vowels present in the English alphabet.

D = {a, e, i, o, u}.

Note: 

Roster Form mostly doesn’t have any element that is repeated

E.g.:

The set of letters which form the word ‘ELEMENT’

E = {E, L, M, N, T}

In this case, no element has got repeated.

  •  Set-builder form 

Talking about the set-builder form, all the elements present in this set have a single common property that is not possessed by any element outside the set.

E.g.:

D = {a, e, i, o, u}

All elements in this set are basically vowels in the English alphabet.

Then, in set-builder form:

D = {x: x is a vowel in English alphabet}

This set would be read as:

“The set of all x such that x is a vowel in the English alphabet.”

C = {y: y is a natural number which divides 42}

B = (z: z is a first four letter of English alphabet}

Types of Sets

Empty Set

A set which does not include any element is called an Empty set. An Empty set is also known as a Null set and Void set. The number of elements in set X is expressed as n(X) and the empty set is signified as Φ. Therefore, n (Φ) = 0. The cardinality of an empty set is zero as it does not have any element.

Singleton Set

A set which is made of only one element is called as a Singleton set. Sometimes, it is called as a unit set and its cardinality is always one.

Finite and Infinite Set

A set which is made of a predetermined number of elements or a finite number of elements is known as a Finite Set. For example: {1, 2, 3, 4, 5, 6} is a finite set with a cardinality of 6 as it has 6 elements.

Union of Sets

The union of two or most of the numbers of sets could be the set of all elements, which belongs to every element of all sets. All the elements are written only once even if they belong to both the sets in the union set of two sets. This is written as ‘∪’. In case, we have set A and B, then the union of these two is A U B and called A union B.

Intersection of Sets

It is basically the set of elements that are common in both the sets. Intersection is a lot like grouping up the common elements. The symbol is denoted as ‘∩’. If A and B are two sets, then the intersection is denoted as A ∩ B and called A intersection B and mathematically. One can write it as:
A∩B={x:x∈A∧x∈B}

Difference of Sets

The difference of set A to B can be written as A – B. So, the set of the element that is in set A not in set B is
A – B = {x: x ∈ A and x ∉ B}

Also, B – A is the set of all elements of the set B which are in B but not in A i.e.
B – A = {x: x ∈ B and x ∉ A}.

Subset of a Set

In the theory of sets, a set P is the subset of any set Q, if the set P is contained in set Q. It simply means that all the elements of the set P belong to the set Q. It is expressed as ‘⊆’ or P ⊆ Q.

Disjoint Sets

When two sets A and B do not possess any common elements or in a situation where the intersection of any two sets A and B is the empty set, then these sets are called disjoint sets i.e. A ∩ B = ϕ. Therefore, the sets are disjoint when this condition n (A ∩ B) = 0 is true.

Equality of Two Sets

Two sets can be called equal or identical to each other when they possess the same elements. When the sets P and Q are equal, and if P ⊆ Q and Q ⊆ P, then we will write it as P = Q.

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