Equal Sets

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1.5 Equal Sets — Study Notes

Definition 3: Equal Sets : Two sets A and B are said to be equal if they contain exactly the same elements.

We write: A = B

If they do not have exactly the same elements, they are called unequal sets.

We write: A ≠ B

For two sets to be equal:

• Every element of A must be in B.


• Every element of B must be in A.


• Order of elements does not matter.


• Repeated elements are counted only once.

Ex 1: Order Does Not Matter

Let: A = {1, 2, 3, 4}, B = {3, 1, 4, 2}

Both sets contain exactly the same elements: 1, 2, 3, 4

Therefore, A = B

Ans: Equal Sets

Ex 2: Prime Numbers

Let A be the set of prime numbers less than 6.

A = {2, 3, 5}

Let P be the set of prime factors of 30.

P = {2, 3, 5}

Both sets contain exactly the same elements.

Therefore,

A = P

Ans: Equal Sets

Note: Repeated Elements

A set does not change if an element is repeated.

For example: A = {1, 2, 3}, B = {2, 2, 1, 3, 3}

After removing repetitions: B = {1, 2, 3}

Therefore, A = B

Note:

In a set: {1, 2, 3} = {1, 1, 2, 2, 3, 3}

Repeated elements do not create new elements.

Points

• Equal Sets: Sets having exactly the same elements.


• Symbol: A = B


• Unequal Sets: Sets that do not have exactly the same elements.


• Symbol: A ≠ B

Trick: i.  Same elements → Equal sets,  ii. Different elements → Unequal sets,  iii. Different order → Still equal,  iv. Repeated elements → Ignore repetition

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Ex 7: Find the Pairs of Equal Sets

Q: Find the pairs of equal sets, if any. Give reasons.

Given:

• 
  
  
  
  • $A=\{0\}$
  
  
  • B = {x : x > 15 and x < 5}
  
  
  • C = {x : x − 5 = 0}
  
  
  • D ={x : x² = 25}
  
  
  • E = {x : x is a positive integral root of x² − 2x − 15 = 0}
  
  
  
  

Solution:

1st : Compare A with the other sets

A = {0}, so 0 ∈ A

But 0 ∉ B, C, D, E

Therefore: A ≠ B, A ≠ C, A ≠ D, A ≠ E

2nd: Find B

For B, we need x > 15 and x < 5 at the same time. This is impossible.

Therefore: B = ∅ 

Since C, D, E are non-empty: B ≠ C, B ≠ D,B ≠ E

3rd: Find C

x − 5 = 0 ⇒ x = 5

Therefore: C = {5}

4th: Find D

x² = 25

x = ± 5

Therefore: D = {−5, 5}

Since −5 ∈ D but  −5 ∉ C : C ≠ D

5th: Find E

x² − 2x − 15 = 0

Factorising: (x−5)(x+3) = 0

So,

x = 5 or x = −3

The question asks for the positive integral root, so only 5 is taken.

Therefore: E = {5}

Hence, C = E

Ans: The only pair of equal sets is: C = E

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Ex 8 : 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 n² ≤ 4} and B = {x : x ∈ R and x² – 3x + 2 = 0}

Solution:

(i) We have, X = {A, L, L, O, Y}, B = {L, O, Y, A, L}

Since repetition of elements does not change a set, we can write: X = {A, L, O, Y}

Also, the order of elements does not matter in a set. Therefore, X = {A, L, O, Y} = B

Hence, X and B are equal sets.

(ii) We have A = {−2, −1, 0, 1, 2}

and

B = {1, 2}

Since 0 ∈ A but 0 ∉ B, the two sets do not have exactly the same elements.

Therefore,

A ≠ B

Hence, A and B are not equal sets.

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Ex 8 : (ii) Which of the following pairs of sets are equal ? Justify your answer.

A = {n : n ∈ Z and n² ≤ 4} and B = {x : x ∈ R and x² – 3x + 2 = 0}

Solution

Given:

A = {n : n ∈ Z and n² ≤ 4}

Since n is an integer,

A = {−2, −1, 0, 1, 2}

Now,

B = {x : x ∈ R and x² − 3x + 2 = 0}

Factorising:

x² − 3x + 2 = 0

(x − 1)(x − 2) = 0

Therefore,

x = 1 or x = 2

So,

B = {1, 2}

Since 0 ∈ A but 0 ∉ B, the two sets do not contain exactly the same elements.

Hence,

A ≠ B

Ans: The two sets are not equal.

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