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

$t\notin B$

So,

$t\in C$

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

Therefore, $A\subseteq \C$

Hence, proved.