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GO Classes DPP | GATE CS | Discrete Mathematics| Set Theory

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Let $A, B$ and $C$ be sets such that $\phi \neq A \cap B \subseteq C$. Then, which of the following statements is not true?

  1. $B \cap C \neq \phi$
  2. If $(A-B) \subseteq C$, then $A \subseteq C$
  3. $(C \cup A) \cap(C \cup B)=C$
  4. If $(A-C) \subseteq B$, then $A \subseteq B$

2 Answers

1 1 vote

 

Given Condition

We are given three sets $A$, $B$, and $C$ such that:

  1. $A \cap B \neq \emptyset$ (Their intersection is not empty).

  2. $A \cap B \subseteq C$ (Every element common to $A$ and $B$ is also in $C$).

Option A: $B \cap C \neq \emptyset$.

Statement: This is True.

Proof:

  • From the given condition, we know there exists at least one element $x$ such that $x \in (A \cap B)$.

  • By the definition of intersection, if $x \in (A \cap B)$, then $x \in A$ and $x \in B$.

  • We are also given $A \cap B \subseteq C$. Therefore, since $x \in (A \cap B)$, it must be that $x \in C$.

  • Since $x \in B$ and $x \in C$, it follows that $x \in (B \cap C)$.

  • Because such an $x$ exists, $B \cap C$ cannot be empty.

Option B: If $(A - B) \subseteq C$, then $A \subseteq C$

Statement: This is True.

Proof:

  • Any set $A$ can be expressed as the union of two disjoint parts: $A = (A \cap B) \cup (A - B)$.

  • We are given:

    • $A \cap B \subseteq C$ (From the problem statement).

    • $A - B \subseteq C$ (From the premise of this option).

  • Taking the union of these two subsets: $(A \cap B) \cup (A - B) \subseteq C \cup C$.

  • Since $(A \cap B) \cup (A - B) = A$ and $C \cup C = C$, we conclude that $A \subseteq C$.

Option C: $(C \cup A) \cap (C \cup B) = C$

Statement: This is True.

Proof:

  • Using the Distributive Law of sets: $(C \cup A) \cap (C \cup B) = C \cup (A \cap B)$.

  • We are given that $A \cap B \subseteq C$.

  • In set theory, if $X \subseteq Y$, then $Y \cup X = Y$.

  • Applying this here with $X = (A \cap B)$ and $Y = C$, we get: $C \cup (A \cap B) = C$.

  • Therefore, the identity holds.

Option D: If $(A - C) \subseteq B$, then $A \subseteq B$

Statement: This is Not True.

Proof by Counter-example:

To prove a statement is not true, we only need one case where the conditions are met but the conclusion fails.

  • Let $A = \{1, 2\}$

  • Let $B = \{1\}$

  • Let $C = \{1, 2\}$

Check Given Conditions:

  1. $A \cap B = \{1\}$. Is it non-empty? Yes.

  2. Is $A \cap B \subseteq C$? $\{1\} \subseteq \{1, 2\}$. Yes.

Check Premise of Option 4:

  • $A - C = \{1, 2\} - \{1, 2\} = \emptyset$.

  • Is $(A - C) \subseteq B$? $\emptyset \subseteq \{1\}$. Yes.

Check Conclusion of Option 4:

  • Is $A \subseteq B$? Is $\{1, 2\} \subseteq \{1\}$? No.

  • The premise is satisfied, but the conclusion is false. Therefore, the statement is not universally true.

Final Answer

The statement that is not true is Option D.

 

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