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# Prove AnB subset A subset AUB?

I shall answer this under the assumption that 'n' means intersection. Recall the definitions of intersection and union: 1) x is an element of AnB if and only if x is an …element of A and x is an element of B 2) x is an element of AUB if and only if x is an element of A or x is an element of B and recall that 3) X is an (improper) subset of Y if and only if every element of X is an element of Y Thus, if x is an element of AnB, then x is an element of A and an element of B, so it clearly is an element A (law of simplification in logic). This implies AnB is a subset of A. Now if x is an element of A, it is certainly an element of A or an element of B (law of addition in logic), and therefore x is an element of AUB. There are other ways of answering this based on axiomatic approaches. ( Full Answer )