Question

IfA= {1,2,3,4), B = {3,4,5,6),
C = {4. 5. 6. 7. 8) and universal set
X= {1.2.3.4.5.6,7,8,9,10), then verify
the following:

2) An(BUC)=(AnB) U (AUC)​

Answer 1

Given

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

The Universal set is:

U = {1,2,3,4,5,6,7,8,9,10}

To Prove

$\sf{A \cap (B \cup C ) = (A \cap B) \cup (A \cap C)}$

LHS

$\sf{A \cap (B \cup C)}$

Firstly,

B U C = {3,4,5,6} U {4,5,........,8}

》 B U C = {3,4,5,6,7,8}

Now,

$\sf{A \cap (B \cup C)}$

》 {1,2,3,4} n {3,4,5,6,7,8}

》 {3,4}................[1]

RHS

$\sf{(A \cap B) \cup (A \cap C)}$

Firstly,

A n B = {3,4}

And, A n C = {4}

Now,

$\sf{(A \cap B) \cup (A \cap C)}$

》 {3,4}............[2]

From equations [1] and [2],we get:

$\sf{A \cap (B \cup C ) = (A \cap B) \cup (A \cap C)}$

Henceforth,proved

Answer 2

Correct Question:

A={1,2,3,4}

B={3,4,5,6}

C={4,5,6,7,8}

Universal set,X={1,2,3,4,5,6,7,8,9,10}

Verify

$\sf A\cap (B\cup C)=(A\cap B)\cup(A\cap C)$

Given:

A={1,2,3,4}

B={3,4,5,6}

C={4,5,6,7,8}

Universal set,X={1,2,3,4,5,6,7,8,9,10}

To Verify:

$\sf A\cap (B\cup C)=(A\cap B)\cup(A\cap C)$

Explanation:

Consider

$\sf B\cup C =(3,4,5,6)\cup (4,5,6,7,8)$

$\sf B\cup C=(3,4,5,6,7,8)$

Consider

$\sf LHS=A\cap (B\cup C)=(1,2,3,4)\cap (3,4,5,6,7,8)$

LHS={3,4}--------(1)

Consider

$\sf A\cap B=(1,2,3,4)\cap (3,4,5,6)=(3,4)$

$\sf A\cap C=(1,2,3,4)\cup (4,5,6,7,8)=(4)$

$\sf RHS=(A\cap B)\cup (A\cap C)$

$\sf RHS=(3,4)\cup (4)$

RHS={3,4}---------(2)

From equation 1 and equation 2

LHS=RHS

Hence proved