Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?
(i) {3, 4} ⊂ A    (ii) {3, 4} ∈ A    (iii) {{3, 4}} ⊂ A    (iv) 1 ∈ A
(v) 1 ⊂ A    (vi) {1, 2, 5} ⊂ A    (vii) {1, 2, 5} ∈ A   (viii) {1, 2, 3} ⊂ A
(ix) φ ∈ A    (x) φ ⊂ A    (xi) {φ} ⊂ A

Solution:

Here are the meanings of ⊂ and ⊄.

If we write A ⊂ B, then we read it as "A is a subset of B" and it means that all the elements of A are present in B.

If we write A ⊄ B, then we read it as "A is NOT a subset of B" and it means that there are some elements of A that are NOT present in B.

It is given that A = {1, 2, {3, 4,}, 5}.

(i) Incorrect as 3 ∈ {3, 4} but 3 ∉ A

(ii) Correct because {3, 4} is an element of A.

(iii) Correct because {3, 4} ∈ {{3, 4}}and {3, 4} ∈ A.

(iv) Correct because 1 is an element of A.

(v) Incorrect because an element of a set can never be a set and hence it cannot be a subset.

(vi) Correct because each element of {1, 2, 5} is also an element of A.

(vii) Incorrect because {1, 2, 5} is not an element of A.

(viii) Incorrect because 3 ∈ {1, 2, 3} ; however, 3 ∉ A.

(ix) Incorrect because f is not an element of A.

(x) Correct because f is a subset of every set.

(xi) Incorrect because of Φ = { } and {Φ} = { {  } } which is NOT present in A.

NCERT Solutions Class 11 Maths Chapter 1 Exercise 1.3 Question 3

Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why? (i) {3, 4} ⊂ A (ii) {3, 4} ∈ A (iii) {{3, 4}} ⊂ A (iv) 1 ∈ A (v) 1 ⊂ A (vi) {1, 2, 5} ⊂ A (vii) {1, 2, 5} ∈ A (viii) {1, 2, 3} ⊂ A (ix) φ ∈ A (x) φ ⊂ A (xi) {φ} ⊂ A

Summary:

For A = { 1, 2, { 3, 4 }, 5 }, we are asked to determine which of the given statements are incorrect and why. We found that 
(i) {3, 4} ⊂ A is incorrect.
(ii) {3, 4} ∈ A is correct.
(iii) {{3, 4}} ⊂ A
(iv) 1 ∈ A is correct.
(v) 1 ⊂ A is incorrect.
(vi) {1, 2, 5} ⊂ A is correct.
(vii) {1, 2, 5} ∈ A is incorrect.
(viii) {1, 2, 3} ⊂ A is incorrect.
(ix) φ ∈ A is incorrect.
(x) φ ⊂ A is correct.
(xi) {φ} ⊂ A is incorrect.