Question

Let A, B and C be sets. Prove that if A ⊆ B ∪ C and A ∩ B = ∅ then A ⊆ C

Answer 1

Answer and Solution:

As per the question:

Given:

If
`A\subseteq B\cup C`

`A\cap B = \phi`

To prove:

`A\subseteq \C`

Proof:

Suppose
`t\in A`

As we know that:

`A\subseteq B\cup C`

Therefore,

`t\in B` or
`t\in C`

Now, if we assume that
`t\in B`

Then

`t\in A\cap B`

Since,

`t\in A` and
`t\in B`

But

A and B are disjoint set and
`A\cap B = \phi`

Therefore, this is contradictory.

Thus

`t\\otin B`

So,

`t\in C`

Every element in the set A is also present in the set C

Therefore,
`A\subseteq \C`

Hence, proved.