Miscellaneous Example (Revised) - Chapter 1 -Sets - Ncert Solutions class 11 - Maths

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Chapter 1 - Sets | NCERT Solutions for Class 11 Maths

Example 1 

Show that the set of letters needed to spell " CATARACT " and the set of letters needed to spell " TRACT" are equal.
Solution

Let $\mathrm{X}$ be the set of letters in "CATARACT". Then
$
\mathrm{X}=\{\mathrm{C}, \mathrm{A}, \mathrm{T}, \mathrm{R}\}
$

Let $\mathrm{Y}$ be the set of letters in "TRACT". Then
$
\mathrm{Y}=\{\mathrm{T}, \mathrm{R}, \mathrm{A}, \mathrm{C}, \mathrm{T}\}=\{\mathrm{T}, \mathrm{R}, \mathrm{A}, \mathrm{C}\}
$

Since every element in $\mathrm{X}$ is in $\mathrm{Y}$ and every element in $\mathrm{Y}$ is in $\mathrm{X}$. It follows that $\mathrm{X}=\mathrm{Y}$.
Example 2

List all the subsets of the set $\{-1,0,1\}$.
Solution

Let $\mathrm{A}=\{-1,0,1\}$. The subset of $\mathrm{A}$ having no element is the empty set $\phi$. The subsets of A having one element are $\{-1\},\{0\},\{1\}$. The subsets of A having two elements are $\{-1,0\},\{-1,1\},\{0,1\}$. The subset of A having three elements of A is A itself. So, all the subsets of A are $\phi,\{-1\},\{0\},\{1\},\{-1,0\},\{-1,1\}$, $\{0,1\}$ and $\{-1,0,1\}$.
Example 3 

Show that $\mathrm{A} \cup \mathrm{B}=\mathrm{A} \cap \mathrm{B}$ implies $\mathrm{A}=\mathrm{B}$
Solution

Let $a \in \mathrm{A}$. Then $a \in \mathrm{A} \cup \mathrm{B}$. Since $\mathrm{A} \cup \mathrm{B}=\mathrm{A} \cap \mathrm{B}, a \in \mathrm{A} \cap \mathrm{B}$. So $a \in \mathrm{B}$.
Therefore, $\mathrm{A} \subset \mathrm{B}$. Similarly, if $b \in \mathrm{B}$, then $b \in \mathrm{A} \cup \mathrm{B}$. Since
$\mathrm{A} \cup \mathrm{B}=\mathrm{A} \cap \mathrm{B}, b \in \mathrm{A} \cap \mathrm{B}$. So, $b \in \mathrm{A}$. Therefore, $\mathrm{B} \subset \mathrm{A}$. Thus, $\mathrm{A}=\mathrm{B}$