Question

If A ⊂ B, what is A ∩ B ?

Answer 1

If x ∈ A ∩ B, then x ∈ A and x ∈ B by definition, so in particular x ∈ A. This proves A ∩ B ⊆ A. Now if x ∈ A, then by assumption x ∈ B, too, so x ∈ A ∩ B. This proves A ⊆ A ∩ B. Together this implies A = A ∩ B.

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Answer 2

Step-by-step explanation:

Some authors use the symbols ⊂ and ⊃ to indicate subset and superset respectively; that is, with the same meaning and instead of the symbols, ⊆ and ⊇.[2] For example, for these authors, it is true of every set A that A ⊂ A.

Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper (also called strict) subset and proper superset respectively; that is, with the same meaning and instead of the symbols, ⊊ and ⊋.[3] This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may or may not equal y, but if x < y, then x definitely does not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if A ⊆ B, then A may or may not equal B, but if A ⊂ B, then A definitely does not equal B.