Question

3. The following sets are not equal - An (BUC) and AU(BnC) Construct a universe U and non-empty sets A, B, and C so that the above sets are in fact the same. . Construct a universe U and non-empty sets A, B, and C so that the above sets are in fact different.

Answer 1

1.U={1,2,3,4,5}

A={2}

B={2,3}

C={4,5}

2.U={1,2,3,4}

A={1,2}

B={2,3}

C={4}

Explanation:

We are given that
`A\cap (B\cup C)` and
`A\cup (B\cap C)`

are different sets

1.We have to construct a universe set U and non empty sets A,B and C so that above set in fact the same

Suppose U={1,2,3,4,5}

A={2}

B={2,3}

C={4,5}

`B\cap C=\phi`

`B\cup C=`{2,3,4,5}

`A\cap (B\cup C)`={2}
`\cap`{2,3,4,5}={2}

`A\cup (B\cap C)`={2}
`\cup\phi`={2}

Hence,
`A\cap (B\cup C)=A\cup (B\cap C)`

2.We have to construct a universe set U and non empty sets A,B and C so that above sets are in fact different

Suppose U={1,2,3,4}

A={1,2}

B={2,3}

C={4}

`B\cap C=\phi`

`B\cup C`={2,3,4}

`A\cup (B\cap C)`={1,2}
`\cup \phi`={1,2}

`A\cap (B\cup C)`={1,2}
`\cap` {2,3,4}={2}

Hence,
`A\cap (B\cup C)\\eq A\cup (B\cap C)`