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 sets C={1,4,5} and D={1,1,4,5,5} are equal because elements of C is in D and 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 x2≤8}, H={x:x∈ℝ and x2-4x+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 x2≤8}
={-2,-1,0,1,2)
[∵x2≤8⇒-2√2≤x≤2√2 and as x∈ℤ ∴x∈{-2,-1,0,1,2}]
and H={x:x∈ℝ and x2-4x+3=0}={1,3}
[∵ x2-4x+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 x2+5x+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 x2+5x+6=0}
First, we find the solution of x2+5x+6=0.
Now, x2+3x+2x+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.
