If \(A=\{x\mid x\in\mathbb{N},\ x^2-4x+3=0\}\) and \(B=\{1\}\), what is the relation between \(A\) and \(B\)?
Answer and explanation
Correct answer: \(B\subset A\) and \(B\neq A\)
Factor the quadratic as \(x^2-4x+3=(x-1)(x-3)\). Thus its natural-number solutions are 1 and 3, so \(A=\{1,3\}\). Since \(B=\{1\}\), every element of B belongs to A, but A has the additional element 3. Therefore B is a proper subset of A: \(B\subset A\) and \(B\neq A\). Option B is false because the sets do not have exactly the same elements, while option D is false because 3 is not in B.
Frequently asked questions
What is the correct answer to this question?
\(B\subset A\) and \(B\neq A\)
Why is this the correct answer?
Factor the quadratic as \(x^2-4x+3=(x-1)(x-3)\). Thus its natural-number solutions are 1 and 3, so \(A=\{1,3\}\). Since \(B=\{1\}\), every element of B belongs to A, but A has the additional element 3. Therefore B is a proper subset of A: \(B\subset A\) and \(B\neq A\). Option B is false because the sets do not have exactly the same elements, while option D is false because 3 is not in B.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Equal sets and Subsets.
