Question

a cross B cross B intersection c equal to a cross B Cross c​

Answer 1

it seems you are being asked to show equality (distributivity of

A×(B∩C)=(A×B)∩(A×C)

and you can do this by "element chasing" I.e., show that both the following hold:

(x,y)∈[A×(B∩C)]⟹(x,y)∈[(A×B)∩(A×C)]

(x,y)∈[(A×B)∩(A×C)]⟹(x,y)∈[A×(B∩C)]

and you're done. But note that you cannot use what you are asked to prove (distributivity of the cross product). Use the definitions of the cross-product, and the definition of set intersection to prove the above (and also distributivity over conjunction/set intersection).

You can also start with unpacking the definition of A×(B∩C) using set-builder notation, and through step by step equivalency, arrive at the set defining (A×B)∩(A×C), showing that we do in fact have that equality holds.

For example:

A×(B∩C)={(x,y)∣x∈A∧y∈(B∩C)}={(x,y)∣x∈A∧(y∈B∧y∈C)}=⋮={(x,y)∣(x∈A∧y∈B)∧(x∈A∧y∈C)}=(A×B)∩(A×C)