How many proper subsets does a set with 3 elements have?

7

A set with n elements has 2ⁿ − 1 proper subsets (excluding the set itself). For n = 3, 2³ − 1 = 7.

Understanding Set Theory questions

Set theory questions deal with collections of objects and how they overlap. A Venn diagram turns nearly every one of them into simple counting.

How to approach it

1. Draw a Venn diagram and fill in the intersection first.
2. Work outwards to the regions belonging to only one set.
3. Apply n(A ∪ B) = n(A) + n(B) − n(A ∩ B) to check totals.
4. Confirm that every element is counted exactly once.

Common mistakes to avoid

• Counting the intersection twice in the union.
• Ignoring elements outside both sets in the universal set.
• Confusing subset with proper subset.

Where it appears: Part of CBSE Class 11 and introductory discrete-maths courses.