**Equality**?

Two sets are equal if they contain exactly the same elements.

Example:

(1) {1,3,4,5} is equal to the set {5,1,4,3}

(2) The set containing the letters of the word railed is equal to the set containing the letters of the word __redial__.

**Equal Sets**

Two sets *A* and *B* are said to be equal, if they have exactly the same elements and we write *A*=*B*. Otherwise, two sets are said to be __unequal__ and we Write either *A*≠*B*.

e.g., Let *A*={a,b,c,d} and *B*={c,d,b,a}, then *A*=*B*, because each element of *A* is in *B* and __vice-versa__.

[

Note]

A set does not change, if one or more elements of the set are repeated. e.g., The setsC={1,4,5} andD={1,1,4,5,5} are equal because elements ofCis inDand vice—versa. That’s why, we generally do not repeat any element in describing a set.

[__Definition__] We say two sets are **equal** if they have exactly the same members.

Example. If

*S*={1,2,3}

then 3∈

*S*and 4∉

*S*. The set membership symbol is often used in defining operations that manipulate sets. The set

*T*={2,3,1}

is equal to

*S*because they have the same members: 1, 2, and 3. While we usually list the members of a set in a “standard” order one is available) there is no requirement to do so and sets are indifferent to the order in which their members are listed.

⛲ Ex1. Show that the set of letters needed to spell “CATARACT” and the set of letters needed to spell “TRACT” are equal.

✍ Solution: Let *X* be the set of letters in “CATARACT”. Then

*X*={C,A,T,R}

Let

*Y*be the set of letters in “TRACT”. Then

*Y*={T,R,A,C,T}={T,R,A,C}

Since every element in

*X*is in

*Y*and every element in

*Y*is in

*X*. It follows that

*X*=

*Y*.

**Equal sets**

Given two sets *A* and *B*, if every elements of *A* is also an element of *B* and if every element of *B* is also an element of *A*, then the sets *A* and *B* are said to be equal. The two equal sets will have exactly the same elements.

⛲ Ex2. Which of the following pairs of sets are equal? Justify your answer.

(i) *E*={*x*:*x* is a letter of the word ”LOYAL”}, *F*={*x*:*x* is a letter of the word ”ALLOY”}

(ii) *G*={*x*:*x*∈ℤ and *x*^{2}≤8}, *H*={*x*:*x*∈ℝ and *x*^{2}-4*x*+3=0}

🌟 First, describe the given sets in the roster form and check whether they have exactly same elements.

✍ Solution:

(i) Given, *E*={*x*:*x* is a letter in the word “LOYAL”}={L,O,Y,A}={A,L,O,Y} and *F*={*x*:*x* is a letter of the word “ALLOY”}={A,L,O,Y}

Here,we see that both sets have exactly the same elements.

*E*=

*F*

(ii) Given,

*G*={

*x*:

*x*∈ℤ and

*x*

^{2}≤8}

[∵

*x*

^{2}≤8⇒-2√2≤

*x*≤2√2 and as

*x*∈ℤ ∴

*x*∈{-2,-1,0,1,2}]

and

*H*={

*x*:

*x*∈ℝ and

*x*

^{2}-4

*x*+3=0}={1,3}

*x*

^{2}-4

*x*+3=0⇒(

*x*-1)(

*x*-3)=0⇒

*x*=1,3]

Here, we see that set

*G*has 5 distinct elements and set

*H*has 2 distinct elements. 50, they do not have same elements.

*G*≠

*H*

Equal Sets

If *A* and *B* are sets such that every element of *A* is an element of *B* and every element of *B* is an element of *A* then *A* and *B* are equal (Identical). We write “*A*=*B*”, and it is read as *A* and *B* are identical.

⛲ Ex3. Is each of the following pair of sets equal? Give reason on each.

(i) *J*={2,3} and *K*={*x*:*x* is a solution of *x*^{2}+5*x*+6=0}

(ii) *M*={*x*:*x* is a letter in the word “FOLLOW”}

and *B*={*y*:*y* is a letter in the word “WOLF”}

🌟 Firstly, convert the given sets in roster form and then check whether they have exactly the same elements.

✍ Solution:

(i) Here, *J*={2,3} and *K*={*x*:*x* is a solution of *x*^{2}+5*x*+6=0}

First, we find the solution of *x*^{2}+5*x*+6=0.

Now, *x*^{2}+3*x*+2*x*+6=0

*x*(

*x*+3)+2(

*x*+3)=0

(

*x*+2)(

*x*+3)=0

*x*=-2, -3

∴

*K*={-2,-3}

Since, the elements of

*J*and

*K*are not same, therefore

*J*≠

*K*.

(ii) Here, *M*={*x*:*x* is a letter of the word “FOLLOW”} ={F,O,L,W}

and *L*={*y*:*y* is a letter of the word “WOLF”}={W,O,L,F}

Since, every element of *M* is in *L* and every element of *L* is in *M* i.e., both have exactly same elements.

*M*=

*L*

🌈 From the Sets given below, Select Equal Sets!