Proposition 18 Let A B C be sets Then A B C A B A C A B C

Proposition 18 Let A, B, C be sets. Then

• A (B C) = (A B) (A C)

1st of proof:

if A (B C) (A B) (A C)

, let x A(B C). Then x A and x B C.

Thus x A and x B or x C.

Hence x A and x B or x A and x C.

In other words, we have x A B or x A C.

Therefore, x (A B) (A C), that is A (B C) (A B) (A C).

then we can say that x (A B) (A C), that is A (B C) = (A B) (A C).proved

2nd of proof:

It is called \"Distributive Property\" for sets.Here is the proof for that,

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)