Four statements are given below regarding elements and subsets of the set {1, 2, {1, 2, 3}}. Only one of them is correct. Which one is it?

Question

Accepted Answer

To determine why Option 2 is the correct answer, let's analyze each option in the context of the set {1, 2, {1, 2, 3}}.

**Option 2: {1, 2} is a proper subset of {1, 2, {1, 2, 3}}**

Explanation:
- A proper subset is a subset that contains some but not all elements of another set. In this case, {1, 2} is indeed a subset of {1, 2, {1, 2, 3}} because both elements 1 and 2 are present in the larger set.
- Additionally, {1, 2} does not contain the element {1, 2, 3}, which is present in the larger set. This makes {1, 2} a proper subset, as it does not include all elements of the set {1, 2, {1, 2, 3}}.
- Therefore, Option 2 is correct.

**Explanation of why the other options are incorrect:**

**Option 1: $\{1,2\}\in\{1,2,\{1,2,3\}\}$**

- The symbol $\in$ denotes membership, meaning the element on the left is a member of the set on the right.
- In this case, {1, 2} is not an element of the set {1, 2, {1, 2, 3}}. The elements of this set are 1, 2, and {1, 2, 3}. The set {1, 2} is not listed as an element, so this statement is incorrect.

**Option 3: {1, 2, 3} is a proper subset of {1, 2, {1, 2, 3}}**

- For {1, 2, 3} to be a proper subset of {1, 2, {1, 2, 3}}, all elements of {1, 2, 3} must be present in the larger set.
- However, the element 3 is not directly present in the set {1, 2, {1, 2, 3}}. Instead, 3 is part of the nested set {1, 2, 3}, which is an element of the larger set.
- Therefore, {1, 2, 3} is not a subset of {1, 2, {1, 2, 3}}, making this statement incorrect.

**Option 4: $3\in\{1,2,\{1,2,3\}\}$**

- The symbol $\in$ indicates that 3 should be a direct element of the set {1, 2, {1, 2, 3}}.
- However, 3 is not directly an element of this set. Instead, it is part of the nested set {1, 2, 3}, which is an element of the larger set.
- Since 3 is not directly listed as an element, this statement is incorrect.

In conclusion, Option 2 is the only statement that correctly describes the relationship between the sets, making it the correct answer.