If A ⊆ B and C ⊆ D, which inclusion must be true?
Answer and explanation
Correct answer: A ∪ C ⊆ B ∪ D
To prove the first inclusion, take any element x in A ∪ C. Then x belongs either to A or to C. If x ∈ A, the condition A ⊆ B gives x ∈ B, so x ∈ B ∪ D. If x ∈ C, the condition C ⊆ D gives x ∈ D, so again x ∈ B ∪ D. Thus every element of A ∪ C is in B ∪ D, proving A ∪ C ⊆ B ∪ D. The reverse and difference/intersection statements are not guaranteed.
Frequently asked questions
What is the correct answer to this question?
A ∪ C ⊆ B ∪ D
Why is this the correct answer?
To prove the first inclusion, take any element x in A ∪ C. Then x belongs either to A or to C. If x ∈ A, the condition A ⊆ B gives x ∈ B, so x ∈ B ∪ D. If x ∈ C, the condition C ⊆ D gives x ∈ D, so again x ∈ B ∪ D. Thus every element of A ∪ C is in B ∪ D, proving A ∪ C ⊆ B ∪ D. The reverse and difference/intersection statements are not guaranteed.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
