If A⊆U and Aᶜ={x∈U: x∉A}, why is A∩Aᶜ empty?
Answer and explanation
Correct answer: Because no element can be both in A and not in A.
By definition, an element belongs to Aᶜ exactly when it belongs to U but does not belong to A. An element in A∩Aᶜ would therefore have to satisfy both x∈A and x∉A at the same time. This is logically impossible, so no element can be common to A and Aᶜ. Consequently, A∩Aᶜ=∅ for every subset A of U, regardless of whether A is empty or equal to U.
Frequently asked questions
What is the correct answer to this question?
Because no element can be both in A and not in A.
Why is this the correct answer?
By definition, an element belongs to Aᶜ exactly when it belongs to U but does not belong to A. An element in A∩Aᶜ would therefore have to satisfy both x∈A and x∉A at the same time. This is logically impossible, so no element can be common to A and Aᶜ. Consequently, A∩Aᶜ=∅ for every subset A of U, regardless of whether A is empty or equal to U.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Complement of a Set and Its Properties.
