GATE Overflow Test Series | Discrete Mathematics | Test 1 | Question: 14

Question

Consider three finite sets $A, B$ and $C.$ Let $S_1 = ((A \cap B) - (B \cap C)) - C$ and $S_2 = (A - (A - C)) - (A -B)$. The sets $S_1$ and $S_2$ are related as.

• A. $S_1 \subset S_2$
• B. $S_1 \supset S_2$
• C. $S_1 = S_2$
• D. None of these

Answer 1

We can use Venn diagram to solve such questions.

$S_1$ and $S_2$ are two disjoint regions in the Venn diagram which can't be compared.

We can also solve them as

$S_1 = ((A \cap B) - (B \cap C)) - C$

$\qquad = ((A \cap B) \cap \overline {(B \cap C)}) - C$

$\qquad = ((A \cap B) \cap (\overline {B} \cup \overline{C})) - C$

$\qquad = (A \cap B \cap  \overline{C}) - C$

$\qquad = (A \cap B \cap  \overline{C}) \cap \overline{C}$

$\qquad = A \cap B \cap  \overline{C}\qquad \to (1)$

$S_2 = (A - (A - C)) - (A -B)$

$\qquad = (A - (A \cap \overline{C}) - (A - B)$

$\qquad = (A \cap  \overline{(A \cap \overline{C})} - (A \cap \overline{B})$

$\qquad = (A \cap  (\overline{A} \cup {C}) - (A \cap \overline{B})$

$\qquad = (A \cap   {C}) - (A \cap \overline{B})$

$\qquad = (A \cap   {C}) \cap  \overline{(A \cap \overline{B})}$

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

$\qquad = (A \cap   {C} \cap    {B})\qquad \to (2)$

$(1)$ and $(2) \implies S_1$ and $S_2$ have no subset relationship.

So, D is the answer.

Comments

Can you please tell me why my approach is wrong for option B

Can i write like option B as A-A+C-A+B=======B+C-A

 

@gatecse @laxshman @shashank

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I am getting option b) for the following set of example:

A={1,2,3}  ; B={2,3,4} ; C={1,4}

S1 = ((A∩B) – (B∩C)) - C

          A∩B = {2,3} and B∩C = {4}

          Now, (A∩B) – (B∩C) = {2,3}    

          and ((A∩B) – (B∩C)) - C = {2,3} – {1,4} = {2,3}

S2 = (A - (A – C)) – (A – B)

          (A – C) = {2,3}

          A - (A – C) = {1}

          (A – B) = {1}

          (A - (A – C)) – (A – B) = {1} – {1} = ∮

As we can see here S2 ⊆ S1 (Hence Option B)

@gatecse can you please confirm this?