If \(A=\{1,2\}\), \(B=\{1,2,3,4,5,6\}\), how many sets \(X\) satisfy \(A\subseteq X\subseteq B\)?
Answer and explanation
Correct answer: 16
Every valid set \(X\) must contain the elements of \(A\), namely 1 and 2, and it cannot contain anything outside \(B\). The optional elements are therefore \(B\setminus A=\{3,4,5,6\}\). Each of these four elements can independently be either included in or excluded from \(X\). Consequently, the number of possible sets is \(2^4=16\), so option C is correct. This is the general formula \(2^{|B|-|A|}\) when \(A\subseteq X\subseteq B\).
Frequently asked questions
What is the correct answer to this question?
16
Why is this the correct answer?
Every valid set \(X\) must contain the elements of \(A\), namely 1 and 2, and it cannot contain anything outside \(B\). The optional elements are therefore \(B\setminus A=\{3,4,5,6\}\). Each of these four elements can independently be either included in or excluded from \(X\). Consequently, the number of possible sets is \(2^4=16\), so option C is correct. This is the general formula \(2^{|B|-|A|}\) when \(A\subseteq X\subseteq B\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
