Let A,B,C be subsets of a set. Prove that, A ∩ B ⊆ C iff A ⊆ Bcompliment ∪ C.

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Given,

A ⊆ B' ∪ C

⟹A ⋂ B ⊆ (B' ⋃ C) ⋂ B (Taking intersection with B on both sides of the equation)

⟹A ⋂ B ⊆ (B' ⋂ B) ⋃ (C ⋂ B) (Distributive property)

⟹A ⋂ B ⊆ ∅ ⋃ (C ⋂ B) (B' ⋂ B = ∅ )

⟹A ⋂ B ⊆ C ⋂ B (∅ ⋃ B = B)

⟹A ⋂ B ⊆ C ( as C ⋂ B ⊆ C )

Hence, proved.