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Q1:
Decide, among the following sets, which sets are subsets of one and another: $A = \{ x : x \in R$ and $x$ satisfy $x^2 – 8x + 12 = 0 \}$, $B = \{ 2, 4, 6 \}$, $C = \{ 2, 4, 6, 8, . . . \}$, $D = \{ 6 \}$.
Solution :
Step 1: Determining the Elements of Set $A$
To identify the elements of set $A$, we solve the quadratic equation $x^2 - 8x + 12 = 0$. [By the Quadratic Formula or Factorization Method].
Factoring the quadratic expression:
$x^2 - 6x - 2x + 12 = 0$
$x(x - 6) - 2(x - 6) = 0$
$(x - 2)(x - 6) = 0$
Setting each factor to zero, we obtain the roots $x = 2$ and $x = 6$.
Thus, $A = \{2, 6\}$.
Step 2: Listing the Sets for Comparison
Based on our calculation and the provided problem statement:
- $A = \{2, 6\}$
- $B = \{2, 4, 6\}$
- $C = \{2, 4, 6, 8, \dots \}$ (The set of all even natural numbers)
- $D = \{6\}$
Step 3: Evaluating Subset Relationships
A set $X$ is a subset of $Y$ ($X \subseteq Y$) if and only if every element of $X$ is also an element of $Y$. [Per the Definition of Subset].
| Relationship | Analysis | Result |
|---|---|---|
| $D \subseteq A$ | $6 \in D$ and $6 \in A$ | True |
| $D \subseteq B$ | $6 \in D$ and $6 \in B$ | True |
| $D \subseteq C$ | $6 \in D$ and $6 \in C$ | True |
| $A \subseteq B$ | $2, 6 \in A$ and $2, 6 \in B$ | True |
| $A \subseteq C$ | $2, 6 \in A$ and $2, 6 \in C$ | True |
| $B \subseteq C$ | $2, 4, 6 \in B$ and $2, 4, 6 \in C$ | True |
Step 4: Geometric Representation of Set Inclusion
Final Solution: The subset relationships are as follows: $D \subset A \subset B \subset C$. Additionally, $D \subset B$, $D \subset C$, and $A \subset C$.
More Questions from Class 11 Mathematics Sets Miscellaneous Exercise on Chapter 1
- 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(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 $A \in C$
- 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 $A \cap B \cap C = \phi$.
CBSE Solutions for Class 11 Mathematics Sets
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