If \(A=\{x\in\mathbb{Z}:x^2-1=0\}\) and \(B=\{x\in\mathbb{Z}:x^2=1\}\), what is \(A-B\)?
Answer and explanation
Correct answer: \(\varnothing\)
To determine A, solve \(x^2-1=0\), which factors as \((x-1)(x+1)=0\). Thus, for integer x, \(x=1\) or \(x=-1\), so \(A=\{-1,1\}\). The condition defining B is already \(x^2=1\), giving the same set \(B=\{-1,1\}\). Since every element of A is also in B, no element remains after subtracting B from A. Hence \(A-B=\varnothing\), so option A is correct.
Frequently asked questions
What is the correct answer to this question?
\(\varnothing\)
Why is this the correct answer?
To determine A, solve \(x^2-1=0\), which factors as \((x-1)(x+1)=0\). Thus, for integer x, \(x=1\) or \(x=-1\), so \(A=\{-1,1\}\). The condition defining B is already \(x^2=1\), giving the same set \(B=\{-1,1\}\). Since every element of A is also in B, no element remains after subtracting B from A. Hence \(A-B=\varnothing\), so option A is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
