The number of triples of sets (A,B,C) with A,B,C⊆1,…,n such that (A∩B)⊆C⊆(A∪B) is:

B. 6^n. Count the choices for one element, then multiply over the n elements. Step 1: In neither A nor B → it cannot be in C ( C⊆A∪B): 1 choice. Step 2: In A only → may or may not be in C: 2 choices. Step 3: In B only → may or may not be in C: 2 choices. Step 4: In both A and B → must be in C ( A∩B⊆C): 1 choice. Step 5: Per element: 1+2+2+1=6 choices, so the total is 6^n. Trap: 8^n ignores the condition (every element free in all three sets).

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The number of triples of sets $(A, B, C)$ with $A, B, C \subseteq 1, \dots, n$ such that $(A \cap B) \subseteq C \subseteq (A \cup B)$ is:

Answer (Detailed Solution Below)View answer ▾

Answer: Option B — $6^{n}$

Detailed Solution

Count the choices for one element, then multiply over the n elements.

In neither $A$ nor $B$ → it cannot be in $C$ ( $C \subseteq A \cup B$ ): 1 choice.

In $A$ only → may or may not be in $C$ : 2 choices.

In $B$ only → may or may not be in $C$ : 2 choices.

In both $A$ and $B$ → must be in $C$ ( $A \cap B \subseteq C$ ): 1 choice.

Per element: $1 + 2 + 2 + 1 = 6$ choices, so the total is $6^{n}$ .

⚠ Common trap

8^{n}

ignores the condition (every element free in all three sets).

Quick tip

Count per element, then raise to the power $n$

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Set Theory — Preparation

This question tests your understanding of Set Theory, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A set is a well-defined collection of objects. Set theory covers types of sets (finite, infinite, empty, singleton, equal and equivalent sets), subsets and power sets, and operations on sets such as union, intersection, difference and complement. It also covers De Morgan's laws and the cardinality formula n(A union B) = n(A) + n(B) - n(A intersection B). In entrance exams, the most common questions are word problems about people who like one, two or three activities, solved with Venn diagrams and the cardinality formula. Questions also ask for the number of subsets of a set (2^n for a set with n elements) and test set notation and properties. Practise the set theory questions below. Each comes with a step-by-step solution and a Venn diagram where it helps.

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The number of triples of sets $(A, B, C)$ with $A, B, C \subseteq 1, \dots, n$ such that $(A \cap B) \subseteq C \subseteq (A \cup B)$ is:

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