Answer to Question #104337 in Algebra for Gayatri Yadav

Question

Let A,B,C be subsets of a set. Prove that A ∩ B ⊆ C iff A ⊆ Bc ∪ C

Answer 1

Method I-

As per the given question,

A and B be a subset of a set C.

Let A= {1,2,3}

B={3,4,5,6,7,8}

C={1,2,3,4,7,8, 9,10}

$A\cap B=\{3,4\}$

So, from the above we can say that $A\cap B \subseteq C$

$B \cup C=\{1,2,3,4,7,8,9,10\}$

So, from the above we can conclude that it is possible if,

$A ⊆ B ∪ C$ ---

Method-II---------------------------------------------------------------------------------------------------------------------------

As per the question,

If A $\subseteq C$ It means all the elements of A is available in the set C.

If $B\subseteq C$ It means all the elements of the B is available in the set C.

Now, $A\cap B$ will give the common elements of the Set A and B.

But, all the elements of A and B are available in the set C,

It means that all the elements of $A\cap B$ must be available in the set C.

So, All the elements of A must be available in the set $B\cup C$

Hence, $A\subseteq B\cup C$

Possibility II

if considering that A,B and C be the subset of the another set E.

Then,

Let A= {1,2,3}

B={3,4,5,6,7,8}

C={3,4,7,8, 9,10}

E={1,2,3,4,5,6,7,8,9,10,11,12}

As per the condition given in the question, A, B, C are the subset of a fourth set E.

$A\cap B$ ={3}-------(i)

But $B\cup C$ ={3,4,5,6,7, 8}-----(ii)

so from the equation (i) and (ii) we can say that A is not the subset of BUC.

Hence the statement given in the question is incorrect.

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Method III