Prove the following distributive law for sets A B C A B C A

Prove the following distributive law for sets A, B, C:

A (B C) = (A B) (A C)

You can use any method you like. For example, you could consider an element x A(BC) and construct a chain of logical deductions to show that x also belongs to (A B) (A C)

Solution

A (B C) = (A B) (A C)

Let x A (B C). If x A (B C) then x is either in A or in (B and C).

x A or x (B and C)

x A or {x B and x C}

{x A or x B} and {x A or x C}

x (A or B) and x (A or C)

x (A B) x (A C)

x (A B) (A C)

x A (B C) => x (A B) (A C)

Therefore,

A (B C) (A B) (A C).........(1)

Let x (A B) (A C). If x (A B) (A C) then x is in (A or B) and x is in (A or C).

x (A or B) and x (A or C)

{x A or x B} and {x A or x C}

x A or {x B and x C}

x A or {x (B and C)}

x A {x (B C)}

x A (B C)

x (A B) (A C) => x A (B C)

Therefore,

(A B) (A C) A (B C)..........(2)

So ,

A (B C) = (A B) (A C)