In 41-13 simple the given expression. Cite a property from Theorem 6.2.2 for every step.

   A ∩ ( ( B ∪ A e ) ∩ B e )

To prove:

A∩((B∪Ac)∩Bc)

Given information

(A∪B)−C=(A−C)∪(B−C)

Concept used:

(A∪B)−C=(A−C)∪(B−C)

Cite a property from Theorem 6.2.2 for every step of the proof.

Solution let A,B and C be any sets. Then

(A∪B)−C=(A∪B)∩Cc               By the set difference law=C​c∩(A∪B)               By the commutative law for ∩=(Cc∩A)∪(Cc∩B)    By the distributive law=(A∩Cc)∪(B∩Cc)    By the commutative law for ∩=(A−C)∪(B−C)         By the set difference law

Calculation:

Consider the expression for the sets A and B written below.

A∩((B∪Ac)∩Bc)

Objective is to simply the above expression. For this consider,

A∩((B∪Ac)∩Bc)

By the distributive law

A∩((B∪Ac)∩Bc)=A∩{(B∩Bc)∪(Ac∩Bc)}

Now use the complementary law