If Aᶜ = {x | x ∈ U and x ∉ A}, why does A ∪ Aᶜ = U hold?
Answer and explanation
Correct answer: Every element of U is either in A or not in A, so it belongs to Aᶜ.
For every element x of U, the law of excluded middle says that either x ∈ A or x ∉ A. If x ∈ A, it is included in A; if x ∉ A, the definition of the complement places it in Aᶜ. Thus every element of U belongs to A or Aᶜ, and both sets are subsets of U. Consequently, A ∪ Aᶜ = U.
Frequently asked questions
What is the correct answer to this question?
Every element of U is either in A or not in A, so it belongs to Aᶜ.
Why is this the correct answer?
For every element x of U, the law of excluded middle says that either x ∈ A or x ∉ A. If x ∈ A, it is included in A; if x ∉ A, the definition of the complement places it in Aᶜ. Thus every element of U belongs to A or Aᶜ, and both sets are subsets of U. Consequently, A ∪ Aᶜ = 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.
