Question

A U (B U C) =(A U B) U C Prove Thst​

Answer 1

hey mate.

let A=(1,2,3)

B=(4,5,6)

C=(7,8,9)

BUC= (4,5,6)U(7,8,9)

=(4,5,6,7,8,9)

AU(BUC)=(1,2,3)U(4,5,6,7,8,9)

(1,2,3,4,5,6,7,8,9)----- equation 1

AUB=(1,2,3) U (4,5,6)

=(1,2,3,4,5,6)

(AUB)UC=(1,2,3,4,5,6)U(7,8,9)

=(1,2,3,4,5,6,7,8,9)------equation 2

From the above sum we can conclude that equation 1 and 2 are equal.

Hence verified

Answer 2

Answer:

To show that two sets are equal, we show inclusions both ways. That is, the first set contains the second and the second set contains the first which can only happen if they are the same set.

AU(BUC) ⊆ (AUB) UC

Let x ∈ AU (BUC)

This means that x∈A or x∈BUC

x ∈A or x ∈ B or x ∈ C = (x ∈A or x ∈ B) or x ∈ C

This is equivalent to x ∈ AUB or x ∈ C.

Therefore, x ∈ (AUB) UC.

Hence it is prove that A U (B U C) =(A U B) U C.

For example,

A = {1,2,3), B = {4,5,6}, C = {7,8,9}

L.H.S= (AUB)uC

= ({1,2,3)u(4,5,6}) u {7,8,9)

= {1,2,3,4,5,6) u {7,8,9}

= {1,2,3,4,5,6,7,8,9)

R.H.S = Au(BUC)

= {1,2,3} u ({4,5,6)u(7,8,9})

= {1,2,3) u (4,5,6,7,8,9}) =

= {1,2,3,4,5,6,7,8,9)

Here L.H.S = R.H.S

So,A U (B U C) =(A U B) U C