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Please answer and show your work. Thumbs up guaranteed.

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For each of the following, determine whether it is valid or invalid. If valid then give a proof. If invalid then give a counter example (A B) (C D) = (A D) (C B) A-(B C) (A-B) (A-C) B C A (C - A) (B-A) is empty (A B) - (A B)= A B is empty

Solution

a> FALSE

EXAMPLE:

suppose,

A = {a,b,c,d} , B = {a, c, d } , C = { a,b,d} , D={d,e}

LHS=>

A int B={a,c,d}

C int D={d}

(A int B) union ( C int D)= {a,c,d} union {d} => {a,c,d}

RHS=>

A int D={d}

C int B={a,d}

(A int D) union ( C int B)= {d} union {a,d} => {a,d}

Hence, LHS != RHS

b> TRUE

EXAMPLE:

A = {a,b,c,d,e} , B = {a, c, d } & C = { a,b,d}

then
LHS=>
B union C = { a,b,c,d }

A - { B union C} = {a,b,c,d,e} - {a,b,c,d} =>{e}

RHS=>

A-B={b,e}

A-C={c,e}

(A-B) int (A-C) ={b,e} int {c,e} =>{e}

Hence, LHS=RHS

c> TRUE

EXAMPLE:

A = {a,b,c,d,e} , B = {a, c, d,f } & C = { a,b,d,e}

B int C={a,d}

and also (b int C) is a subset of A.

C-A={e}

B-A={f}

(C-A) int (B-A) ={e} int {f} =>{ } , which results in empty set. Hence Proved.

d> TRUE

EXAMPLE:

suppose,

A = {a,b} , B = { } Where, B is empty set.

A union B ={a,b,}

A int B={}

(A union B )-(A int B)={a,b,}-{} => {a,b} which is nothing but set A when B is empty set.