Difference Between Equal Set and Equivalent Set – Class 11 Applied Mathematics

Definition of Equal Sets

Two sets A and B are said to be equal if they contain exactly the same elements. The order of elements does not matter, but every element in A must be in B and vice versa. We denote this as A = B.

A = B ⟺ (∀x ∈ A ⟹ x ∈ B) ∧ (∀y ∈ B ⟹ y ∈ A)

Example: Solved Example: Verify if A = {1, 2, 3} and B = {3, 1, 2} are equal.

Show Step-by-Step Solution

Step 1: List elements of A: {1, 2, 3}.
Step 2: List elements of B: {3, 1, 2}.
Step 3: Compare elements: 1, 2, and 3 are present in both sets.
Answer: Since all elements are identical, A = B.

Definition of Equivalent Sets

Two sets A and B are equivalent if they have the same number of elements, known as cardinality. This is denoted by n(A) = n(B). Equivalent sets do not need to share the same elements.

A ≈ B ⟺ n(A) = n(B)

Example: Solved Example: Check if A = {a, b, c} and B = {1, 2, 3} are equivalent.

Show Step-by-Step Solution

Step 1: Count elements in A: n(A) = 3.
Step 2: Count elements in B: n(B) = 3.
Step 3: Compare cardinalities: 3 = 3.
Answer: Since n(A) = n(B), the sets are equivalent.

Key Differences and Relationships

Every equal set is necessarily equivalent because they share the same cardinality. However, an equivalent set is not necessarily equal because the elements themselves may differ. Equal sets are a subset of equivalent sets.

(A = B) ⟹ (A ≈ B) but (A ≈ B) ⇏ (A = B)

Example: Solved Example: Compare A = {1, 2} and B = {3, 4}.

Show Step-by-Step Solution

Step 1: Check equality: {1, 2} ≠ {3, 4}.
Step 2: Check cardinality: n(A) = 2, n(B) = 2.
Step 3: Conclusion: Sets are equivalent but not equal.

Section A: Multiple Choice Questions (MCQs)

easyIf A = {5, 10, 15} and B = {15, 5, 10}, which statement is true?

A. A ≈ B but A ≠ B B. A = B C. A ≠ B and A ≉ B D. n(A) ≠ n(B)

Show Answer

✔ Correct: B

Both sets contain the exact same elements {5, 10, 15}, therefore A = B.

easyIf A = {x : x is a natural number ≤ 3} and B = {1, 2, 3}, then:

A. A ≈ B B. A ≠ B C. A = B D. Both A and C

Show Answer

✔ Correct: D

A = {1, 2, 3}. Since A and B have same elements, they are equal, and all equal sets are equivalent.

mediumGiven A = {a, e, i, o, u} and B = {1, 2, 3, 4, 5}, which is correct?

A. A = B B. A ≈ B C. A ≉ B D. n(A) ≠ n(B)

Show Answer

✔ Correct: B

n(A) = 5 and n(B) = 5. Since cardinalities are equal, they are equivalent, but elements differ.

mediumIf A = {x : x² - 3x + 2 = 0} and B = {1, 2}, then:

A. A = B B. A ≈ B C. A ≠ B D. Both A and B

Show Answer

✔ Correct: D

Solving x² - 3x + 2 = 0 gives x = 1, 2. So A = {1, 2}. Since A = B, they are also equivalent.

hardLet A = {x : x is a prime number < 10} and B = {2, 3, 5, 7}. Which is true?

A. A = B B. A ≈ B C. A ≠ B D. Both A and B

Show Answer

✔ Correct: D

A = {2, 3, 5, 7}. Since A = B, they are equal and equivalent.

hardIf A = {1, 2, 3} and B = {x, y, z}, then A and B are:

A. Equal B. Equivalent C. Neither D. Equal and Equivalent

Show Answer

✔ Correct: B

n(A) = 3 and n(B) = 3. They are equivalent because cardinalities match, but not equal as elements differ.

very hardIf A = {x : x ∈ N, x < 5} and B = {x : x ∈ Z, x² = 1, 4, 9, 16}, then:

A. A = B B. A ≈ B C. A ≠ B and A ≉ B D. A ≉ B

Show Answer

✔ Correct: B

A = {1, 2, 3, 4} (n=4). B = {1, -1, 2, -2, 3, -3, 4, -4} (n=8). Wait, checking B: x² = 1, 4, 9, 16 implies x = ±1, ±2, ±3, ±4. n(B)=8. A ≈ B is false. Correct answer is A ≠ B and A ≉ B (C).

Section B: Assertion–Reason Questions

Assertion (A): If A = {1, 2} and B = {2, 1}, then A = B.
Reason (R): Equal sets must contain the exact same elements regardless of order.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: A

Both assertion and reason are true, and the reason correctly explains the equality.

Assertion (A): If A = {1, 2, 3} and B = {4, 5, 6}, then A = B.
Reason (R): Equivalent sets have the same number of elements.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: D

Assertion is false (elements differ). Reason is true (definition of equivalent sets).

Section C: Short Answer Questions

mediumGiven A = {x : x is a letter in 'APPLE'} and B = {x : x is a letter in 'ALEAP'}, find: (i) Elements of A (ii) Elements of B (iii) Are they equal or equivalent?

Show Answer & Solution

(i) A = {A, P, L, E}
(ii) B = {A, L, E, P}
(iii) Since A = B, they are both equal and equivalent.

mediumGiven A = {1, 3, 5}, B = {2, 4, 6}, C = {5, 1, 3}. Find: (i) n(A) and n(B) (ii) Is A ≈ B? (iii) Is A = C?

Show Answer & Solution

(i) n(A) = 3, n(B) = 3
(ii) Yes, because n(A) = n(B)
(iii) Yes, because they contain the same elements {1, 3, 5}.

hardGiven A = {x : x ∈ N, x² ≤ 9}, B = {1, 2, 3}, C = {3, 2, 1}. Find: (i) Set A (ii) Is A = B? (iii) Is B = C?

Show Answer & Solution

(i) A = {1, 2, 3}
(ii) Yes, A = {1, 2, 3} and B = {1, 2, 3}, so A = B
(iii) Yes, B = {1, 2, 3} and C = {1, 2, 3}, so B = C.

Section D: Long Answer Questions

hardLet A = {x : x is an even prime number}, B = {x : x ∈ N, x < 3 and x > 1}, and C = {2}. Prove that A = B = C and determine if they are equivalent.

Show Answer & Solution

Step 1: Identify elements of A: The only even prime is 2. So A = {2}.
Step 2: Identify elements of B: Natural numbers between 1 and 3 is 2. So B = {2}.
Step 3: Identify elements of C: C = {2}.
Step 4: Compare: A = {2}, B = {2}, C = {2}. Since all elements are identical, A = B = C.
Step 5: Cardinality: n(A)=1, n(B)=1, n(C)=1. Since n(A)=n(B)=n(C), they are equivalent.

very hardConsider set A = {x : x is a root of x² - 5x + 6 = 0} and B = {2, 3}. Let C = {x : x ∈ N, x < 3}. Verify if A = B and if A ≈ C.

Show Answer & Solution

Step 1: Solve x² - 5x + 6 = 0: (x-2)(x-3) = 0, so x = 2, 3. A = {2, 3}.
Step 2: Compare A and B: A = {2, 3}, B = {2, 3}. Since elements are identical, A = B.
Step 3: Identify C: Natural numbers < 3 are {1, 2}. C = {1, 2}.
Step 4: Check A ≈ C: n(A) = 2, n(C) = 2. Since 2 = 2, A ≈ C.
Step 5: Final Conclusion: A = B is true, and A ≈ C is true.

Section E: Case-Based Question

★

Case Study: A library has two shelves. Shelf A contains books on {Maths, Physics, Chemistry}. Shelf B contains books on {History, Geography, Civics}. Both shelves have 3 books each.

Sub-question 1: (i) Are the sets of books on Shelf A and Shelf B equal? (ii) Are they equivalent?

Show Solution

(i) No, the subjects are different. (ii) Yes, n(A) = 3 and n(B) = 3, so they are equivalent.

Sub-question 2: If Shelf C is added with {Physics, Maths, Chemistry}, compare Shelf A and Shelf C.

Show Solution

Shelf A = {Maths, Physics, Chemistry}, Shelf C = {Physics, Maths, Chemistry}. They are equal (A = C) and therefore also equivalent.

CBSE Board & Sample Paper Past Year Questions

📋 CBSE 2023 📝 1 Mark

If A = {1, 2} and B = {1, 2, 3}, are they equivalent?

Show Answer & Solution

No. n(A) = 2 and n(B) = 3. Since 2 ≠ 3, they are not equivalent.

📋 CBSE 2022 📝 2 Marks

Define equal sets with an example.

Show Answer & Solution

Two sets are equal if they contain the same elements. Example: A = {a, b}, B = {b, a}. A = B.

📋 CBSE Sample Paper 2024 📝 1 Mark

If A = {x : x is a digit in 121}, B = {1, 2}. Are they equal?

Show Answer & Solution

✔ Answer: A

A = {1, 2}. B = {1, 2}. Yes, they are equal.

📋 CBSE 2020 📝 1 Mark

If A = {a, b, c} and B = {1, 2, 3}, are they equivalent?

Show Answer & Solution

✔ Answer: A

Yes, n(A) = 3 and n(B) = 3. Since cardinalities are equal, they are equivalent.

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