Mathematics Sets and their Representations,

Sets and their Representations :

- A set is a well-defined collection of objects.

- If we want to find out the collection of five most renowned mathematicians of the world , then it is not well-defined, because the criterion for determining a mathematician as most renowned may vary from person to person. So this can't be a set.

- If `a` is an element of a set `A,` we say that “ a belongs to A” the Greek symbol ∈ (epsilon) is used to denote the phrase ‘belongs to’. Thus, we write `a ∈ A`. If ‘b’ is not an element of a set A, we write `b ∉ A` and read “b does not belong to A.

-Some Examples of set

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, and
R+ : the set of positive real numbers.

-The following points may be noted :

(i) If same element comes twice or more time in set, then its consider once at final set.
(ii) Sets are usually denoted by capital letters `A, B, C, X, Y, Z`, etc.
(iii) The elements of a set are represented by small letters `a, b, c, x, y, z`, etc.

Representation of a Set :

There are two methods of representing a set :

(i) Roster or tabular form
(ii) Set-builder form

-(I)`" Roster or tabular form :"`

In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces `{ }.`

E.g. the set of all even positive integers less than 7 is described in roster form as {2, 4, 6}.

`"(ii) Set-builder form :"`

- In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.

Method : - We describe the element of the set by using a symbol `x,` which is followed by a colon “ : ”.
- After the sign of colon, we write the characteristic property possessed by the elements of the set and then enclose the whole description within braces.

E.g. In the set `{a, e, i, o, u}`, is a vowel in the English alphabet. Denoting this set by V, we can write as `V = {x : x \ \ "is a vowel in English alphabet}"` in set builder form.

E.g. A = {x : x is a natural number and 3 < x < 10} is read as “the set of all x such that `x` is a natural number and x lies between 3 and 10.

Hence, the numbers 4, 5, 6, 7, 8 and 9 are the elements of the set A.

Q 2636334272

Write the set `A = {1, 4, 9, 16, 25, . . . }` in set-builder form.

Solution:

We may write the set `A` as

`A = `{ `x : x` is the square of a natural number}

Alternatively, we can write

`A = {x : x = n^2, "where " n ∈ N}`

Q 2616334270

Write the set {`x : x` is a positive integer and `x^2 < 40` } in the roster form.

Solution:

The required numbers are `1, 2, 3, 4, 5, 6`. So, the given set in the roster form is `{1, 2, 3, 4, 5, 6}`.

The Empty Set

Definition : A set which does not contain any element is called the empty set or the null set or the void set, is denoted by the symbol `φ` or { }.

E.g.

(i) Let `A =` {`x : 1 < x < 2, x` is a natural number}. Here is no natural number between 1 and 2. So `A` is the empty set.

(ii) `B =` {`x : x^2 – 2 = 0` and `x` is rational number}. Then `B` is the empty set because the equation `x^2 – 2 = 0` is not satisfied by any rational value of `x`.

Finite and Infinite Sets & Cardinality of a Finite Set

Definition : A set which is empty or consists of a definite number of elements is called finite otherwise, the set is called infinite.

Consider some examples :

(i) Let W be the set of the days of the week. Then W is finite.
(ii) Let S be the set of solutions of the equation `x^2 –16 = 0`. Then `S` is finite.

(iii) Let G be the set of points on a line. Then G is infinite.

Note : All infinite sets cannot be described in the roster form. For example, the set of real numbers cannot be described in this form, because the elements of this set do not follow any particular pattern.

`"Cardinality"` of a finite Set is defined as total number of elements in a set.

Q 2646334273

State which of the following sets are finite or infinite :
(i) {`x : x ∈ N` and `(x – 1) (x –2) = 0}`
(ii) {`x : x ∈ N` and `x^2 = 4`}
(iii) {`x : x ∈ N` and `2x –1 = 0`}
(iv) {`x : x ∈ N` and `x` is prime}
(v) {`x : x ∈ N` and `x` is odd}

Solution:

(i) Given set `= {1, 2}`. Hence, it is finite.
(ii) Given set `= {2}`. Hence, it is finite.
(iii) Given set `= φ`. Hence, it is finite.
(iv) The given set is the set of all prime numbers and since set of prime numbers is infinite. Hence the given set is infinite
(v) Since there are infinite number of odd numbers, hence, the given set is infinite.

Equal Sets :

Definition : Two sets A and B are said to be equal if they have exactly the same elements and we write A = B. Otherwise, the sets are said to be unequal and we write `A ≠ B`.

We consider the following examples :

(i) Let `A = {1, 2, 3, 4}` and `B = {3, 1, 4, 2}`. Then `A = B`.

(ii) Let `A` be the set of prime numbers less than 6 and P the set of prime factors of `30`. Then A and P are equal, since 2, 3 and 5 are the only prime factors of `30` and also these are less than `6`.

Note : A set does not change if one or more elements of the set are repeated. For example, the sets `A = {1, 2, 3}` and `B = {2, 2, 1, 3, 3}` are equal, since each element of A is in B and vice-versa .

Q 2686334277

Which of the following pairs of sets are equal? Justify your answer.
(i) X, the set of letters in “ALLOY” and B, the set of letters in “LOYAL”.
(ii) `A =` {`n : n ∈ Z` and `n^2 ≤ 4`} and `B =` {`x : x ∈ R` and `x^2 – 3x + 2 = 0`}.

Solution:

(i) We have, `X = {A, L, L, O, Y}, B = {L, O, Y, A, L}`. Then X and B are equal sets as repetition of elements in a set do not change a set. Thus,

`X = {A, L, O, Y} = B`

(ii) `A = {–2, –1, 0, 1, 2}, B = {1, 2}`. Since `0 ∈ A` and `0 ∉ B, A` and `B` are not equal sets.

Subsets :

- Definition : A set A is said to be a subset of a set B if every element of A is also an element of B.

- We write as, `A subseteq B,` and is called A to be a subset of B, all that is needed is that every element of A is in B. It is possible that every element of B may or may not be in A.

E.g. `A subseteq B` if `a ∈ A ⇒ a ∈ B`

- We read the above statement as “A is a subset of B if `a` is an element of `A` implies that `a` is also an element of `B`”. If A is not a subset of B, we write `A ⊈ B`.

- If Every element of B is also in A and if every element of A may or may not be in B. In this case, A and B are the same sets so that we have `A subseteq B` and `B subseteq A ⇔ A = B`, where `“⇔”` is a symbol for two way implications, and is usually read as if and only if .

Note : It follows from the above definition that every set A is a subset of itself, i.e., `A subseteq A`. Since the empty set `φ` has no elements, we agree to say that `φ` is a subset of every set.

Q 2616034879

List all the subsets of the set `{ –1, 0, 1 }`.

Solution:

Let `A = { –1, 0, 1 }`. The subset of A having no element is the empty set `φ`. The subsets of A having one element are `{ –1 }, { 0 }, { 1 }`. The subsets of A having two elements are `{–1, 0}, {–1, 1} ,{0, 1}`. The subset of A having three elements of A is A itself. So, all the subsets of A are `φ, {–1}, {0}, {1}, {–1, 0}, {–1, 1}, {0, 1}` and `{–1, 0, 1}`.

Power Set :

Definition : The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set.
E.g. If `A = { 1, 2 }`, then `P( A ) = { φ,{ 1 }, { 2 }, { 1,2 }}` Also, note that `n [ P (A) ] = 4 = 2^2`

In general, if `A` is a set with `n(A) = m`, then it can be shown that `n [ P(A)] = 2^m`

Universal Set :

- Universal Set is the set defines as the set containing all objects or elements and of which all other sets are subsets.

or

- A universal set is the collection of all objects in a particular context or theory. All other sets in that framework constitute subsets of the universal set.

- Universal Sets are usually named with a capital letter. Therefore, the universal set is usually named with the capital letter `U.`

For example, R can be set of real numbers.

Examples :

Sets and their Representations, - Basics and Definitions