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Q2(ii):
In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example. (ii) If $A \subset B$ and $B \in C$, then $A \in C$
Solution :
Step 1: Analyzing the Logical Structure
The statement is: If $A \subset B$ and $B \in C$, then $A \in C$.
To evaluate this, we define the relations as follows:
- $A \subset B$: Every element $x$ such that $x \in A$ implies $x \in B$.
- $B \in C$: The set $B$ is an element of the set $C$.
- Conclusion: $A \in C$: The set $A$ must be an element of the set $C$.
[Per the definitions of Subset and Elementhood in Set Theory, the relation of "being an element of" is not necessarily transitive across the relation of "being a subset of".]
Step 2: Determining Truth Value
The statement is False. Inclusion ($A \subset B$) defines a relationship between sets, while elementhood ($B \in C$) defines membership of a set within a collection. Having $B$ as a member of $C$ does not imply that every subset of $B$ is also a member of $C$.
Step 3: Providing a Counter-Example
To disprove the universal quantification of this statement, we need only one counter-example where the premises are true but the conclusion is false.
| Set | Example Definition |
|---|---|
| $A$ | $\{1\}$ |
| $B$ | $\{1, 2\}$ |
| $C$ | $\{\{1, 2\}\}$ |
Verification of Premises:
- $A \subset B$: Since $1 \in \{1\}$ and $1 \in \{1, 2\}$, the inclusion holds. [By Subset Definition].
- $B \in C$: The set $\{1, 2\}$ is indeed the unique element of set $C$. The premise holds.
Verification of Conclusion:
- $A \in C$: Is $\{1\} \in \{\{1, 2\}\}$? No. The only element in $C$ is $\{1, 2\}$. Therefore, $A \notin C$.
Step 4: Visualizing the Membership Hierarchy
Final Solution: The statement is False. As shown by the counter-example where $A=\{1\}$, $B=\{1, 2\}$, and $C=\{\{1, 2\}\}$, it is observed that while $A \subset B$ and $B \in C$, it does not follow that $A \in C$.
More Questions from Class 11 Mathematics Sets Miscellaneous Exercise on Chapter 1
- Q1: Decide, among the following sets, which sets are subsets of one and another: $A = \{ x : x \in R$ and $x$ satisfy $x^2 – 8x + 12 = 0 \}$, $B = \{ 2, 4, 6 \}$, $C = \{ 2, 4, 6, 8, . . . \}$, $D = \{ 6 \}$.
- Q2(i): In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example. (i) If $x \in A$ and $A \in B$, then $x \in B$
- Q2(iii): In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example. (iii) If $A \subset B$ and $B \subset C$, then $A \subset C$
- Q2(iv): In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example. (iv) If $A \not\subset B$ and $B \not\subset C$, then $A \not\subset C$
- Q2(v): In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example. (v) If $x \in A$ and $A \not\subset B$, then $x \in B$
- Q2(vi): In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example. (vi) If $A \subset B$ and $x \notin B$, then $x \notin A$
- Q3: Let $A, B$, and $C$ be the sets such that $A \cup B = A \cup C$ and $A \cap B = A \cap C$. Show that $B = C$.
- Q4(i): Show that the following four conditions are equivalent : (i) $A \subset B$
- Q4(ii): Show that the following four conditions are equivalent : (ii) $A – B = \phi$
- Q4(iii): Show that the following four conditions are equivalent : (iii) $A \cup B = B$
- Q4(iv): Show that the following four conditions are equivalent : (iv) $A \cap B = A$
- Q5: Show that if $A \subset B$, then $C – B \subset C – A$.
- Q6: Show that for any sets $A$ and $B$, $A = ( A \cap B ) \cup ( A – B )$ and $A \cup ( B – A ) = ( A \cup B )$
- Q7(i): Using properties of sets, show that (i) $A \cup ( A \cap B ) = A$
- Q7(ii): Using properties of sets, show that (ii) $A \cap ( A \cup B ) = A$.
- Q8: Show that $A \cap B = A \cap C$ need not imply $B = C$.
- Q9: Let $A$ and $B$ be sets. If $A \cap X = B \cap X = \phi$ and $A \cup X = B \cup X$ for some set $X$, show that $A = B$. (Hints $A = A \cap ( A \cup X ) , B = B \cap ( B \cup X )$ and use Distributive law )
- Q10: Find sets $A, B$ and $C$ such that $A \cap B, B \cap C$ and $A \cap C$ are non-empty sets and $A \cap B \cap C = \phi$.
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