If \(A=\{1,2,3\}\), how many elements does \(\mathcal{P}(A)\setminus\{\varnothing,A\}\) contain?
Answer and explanation
Correct answer: 6
A set with three elements has \(2^3=8\) subsets, so \(|\mathcal{P}(A)|=8\). The set difference removes two specific elements of the power set: the empty set \(\varnothing\) and the complete set \(A\) itself. These are distinct subsets, leaving \(8-2=6\) elements. Equivalently, the remaining subsets are the nonempty proper subsets of A. Hence option C is correct.
Frequently asked questions
What is the correct answer to this question?
6
Why is this the correct answer?
A set with three elements has \(2^3=8\) subsets, so \(|\mathcal{P}(A)|=8\). The set difference removes two specific elements of the power set: the empty set \(\varnothing\) and the complete set \(A\) itself. These are distinct subsets, leaving \(8-2=6\) elements. Equivalently, the remaining subsets are the nonempty proper subsets of A. Hence option C is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
