Sets

(A set is a subset of another set if every element of the first set is also an element of the second.)

(i) $\{2,3,4\}\ldots\{1,2,3,4,5\}$

Sol. Every element of the set $\{2,3,4\}$ is also an element of the set $\{1,2,3,4,5\}$.

∴ $\{2,3,4\}\subset\{1,2,3,4,5\}$

(ii) $\{a,b,c\}\ldots\{b,c,d\}$

Sol. $a\in\{a,b,c\}$ $\text{but }a\notin\{b,c,d\}$.

∴ $\{a,b,c\}\not\subset\{b,c,d\}$

(iii) $\{x:x$ is a student of Class XI of your $\text{school}\}\ldots\{x:x$ student of your $\text{school}\}$

Sol. Every student of Class XI of your school is a student of your school.

∴ $\{x:x$ is a student of Class XI of your $\text{school}\}\subset\{x:x$ student of your $\text{school}\}$

(iv) $\{x:x$ is a circle in the $\text{plane}\}\ldots\{x:x$ is a circle in the same plane with radius 1 $\text{unit}\}$

Sol. A circle of radius $2$ units in the plane is an element of the first set but not of the second.

∴ $\{x:x$ is a circle in the $\text{plane}\}\not\subset\{x:x$ is a circle in the same plane with radius 1 $\text{unit}\}$

(v) $\{x:x$ is a triangle in a $\text{plane}\}\ldots\{x:x$ is a rectangle in the $\text{plane}\}$

Sol. A triangle is not a rectangle.

∴ $\{x:x$ is a triangle in a $\text{plane}\}\not\subset\{x:x$ is a rectangle in the $\text{plane}\}$

(vi) $\{x:x$ is an equilateral triangle in a $\text{plane}\}\ldots\{x:x$ is a triangle in the same $\text{plane}\}$

Sol. Every equilateral triangle is a triangle.

∴ $\{x:x$ is an equilateral triangle in a $\text{plane}\}\subset\{x:x$ is a triangle in the same $\text{plane}\}$

(vii) $\{x:x$ is an even natural $\text{number}\}\ldots\{x:x$ is an $\text{integer}\}$

Sol. Every even natural number is an integer.

∴ $\{x:x$ is an even natural $\text{number}\}\subset\{x:x$ is an $\text{integer}\}$

(i) $\{a,b\}\not\subset\{b,c,a\}$

Sol. Both $a$ and $b$ are elements of $\{b,c,a\}$, so $\{a,b\}\subset\{b,c,a\}$.

∴ The statement is false.

(ii) $\{a,e\}\subset\{x:x$ is a vowel in the English $\text{alphabet}\}$

Sol. The set of vowels in the English alphabet is $\{a,e,i,o,u\}$.

Every element of $\{a,e\}$ is also an element of $\{a,e,i,o,u\}$, so $\{a,e\}\subset\{a,e,i,o,u\}$.

∴ The statement is true.

(iii) $\{1,2,3\}\subset\{1,3,5\}$

Sol. $2\in\{1,2,3\}$ $\text{but }2\notin\{1,3,5\}$.

∴ The statement is false.

(iv) $\{a\}\subset\{a,b,c\}$

Sol. The only element $a$ of $\{a\}$ is an element of $\{a,b,c\}$.

∴ The statement is true.

(v) $\{a\}\in\{a,b,c\}$

Sol. The elements of $\{a,b,c\}$ are $a,b,c$; the set $\{a\}$ is not one of them (the correct statements $\text{are }a\in\{a,b,c\}$ and $\{a\}\subset\{a,b,c\}$).

∴ The statement is false.

(vi) $\{x:x$ is an even natural number less than $\text{6}\}\subset\{x:x$ is a natural number which divides $\text{36}\}$

Sol. The even natural numbers less than $6$ are $2,4$, so the first set is $\{2,4\}$.

The natural numbers which divide $36$ are $1,2,3,4,6,9,12,18,36$, so the second set is $\{1,2,3,4,6,9,12,18,36\}$.

Both $2$ and $4$ are in the second set.

∴ The statement is true.

(The elements of $A$ are $1$, $2$, $\{3,4\}$ and $5$. Note that $3$ and $4$ by themselves are not elements of $A$.)

(i) $\{3,4\}\subset A$

Sol. $3\in\{3,4\}$ but $3\notin A$.

∴ $\{3,4\}\subset A$ is incorrect.

(ii) $\{3,4\}\in A$

Sol. $\{3,4\}$ is an element of $A$.

∴ $\{3,4\}\in A$ is correct.

(iii) $\{\{3,4\}\}\subset A$

Sol. The only element of $\{\{3,4\}\}$ is $\{3,4\}$, which is an element of $A$.

∴ $\{\{3,4\}\}\subset A$ is correct.

(iv) $1\in A$

Sol. $1$ is an element of $A$.

∴ $1\in A$ is correct.

(v) $1\subset A$

Sol. $1$ is an element, not a set, so the symbol $\subset$ cannot be used between $1$ and $A$ (the correct forms are $1\in A$ or $\{1\}\subset A$).

∴ $1\subset A$ is incorrect.

(vi) $\{1,2,5\}\subset A$

Sol. Each of $1,2,5$ is an element of $A$.

∴ $\{1,2,5\}\subset A$ is correct.

(vii) $\{1,2,5\}\in A$

Sol. $\{1,2,5\}$ is not one of the four elements of $A$.

∴ $\{1,2,5\}\in A$ is incorrect.

(viii) $\{1,2,3\}\subset A$

Sol. $3\in\{1,2,3\}$ but $3\notin A$.

∴ $\{1,2,3\}\subset A$ is incorrect.

(ix) $\phi\in A$

Sol. $\phi$ is not one of the elements of $A$.

∴ $\phi\in A$ is incorrect.

(x) $\phi\subset A$

Sol. $\phi$ is a subset of every set.

∴ $\phi\subset A$ is correct.

(xi) $\{\phi\}\subset A$

Sol. The only element of $\{\phi\}$ is $\phi$, and $\phi\notin A$.

∴ $\{\phi\}\subset A$ is incorrect.

Hence, the incorrect statements are (i), (v), (vii), (viii), (ix) and (xi).

(i) $\{a\}$

(ii) $\{a,b\}$

(iii) $\{1,2,3\}$

(iv) $\phi$

(i) The subsets of $\{a\}$ are $\varnothing$ and $\{a\}$.

(ii) The subsets of $\{a,b\}$ are $\varnothing$, $\{a\}$, $\{b\}$ and $\{a,b\}$.

(iii) The subsets of $\{1,2,3\}$ are $\varnothing$, $\{1\}$, $\{2\}$, $\{3\}$, $\{1,2\}$, $\{1,3\}$, $\{2,3\}$ and $\{1,2,3\}$.

(iv) The only subset of $\varnothing$ is $\varnothing$.

(i) $\{x:x\in R,\ -4<x\le6\}$

(ii) $\{x:x\in R,\ -12<x<-10\}$

(iii) $\{x:x\in R,\ 0\le x<7\}$

(iv) $\{x:x\in R,\ 3\le x\le4\}$

(i) $\{x:x\in\mathbb{R},\;-4<x\le6\}=(-4,6]$

(ii) $\{x:x\in\mathbb{R},\;-12<x<-10\}=(-12,-10)$

(iii) $\{x:x\in\mathbb{R},\;0\le x<7\}=[0,7)$

(iv) $\{x:x\in\mathbb{R},\;3\le x\le4\}=[3,4]$

(i) $(-3,0)$

(ii) $[6,12]$

(iii) $(6,12]$

(iv) $[-23,5)$

(i) $(-3,0)=\{x:x\in\mathbb{R},\;-3<x<0\}$

(ii) $[6,12]=\{x:x\in\mathbb{R},\;6\le x\le12\}$

(iii) $(6,12]=\{x:x\in\mathbb{R},\;6<x\le12\}$

(iv) $[-23,5)=\{x:x\in\mathbb{R},\;-23\le x<5\}$

(i) The set of right triangles.

Sol. For the set of right triangles, the universal set can be the set of all triangles or the set of all polygons.

(ii) The set of isosceles triangles.

Sol. For the set of isosceles triangles, the universal set can be the set of all triangles, the set of all polygons or the set of all two-dimensional figures.

(i) $\{0,1,2,3,4,5,6\}$

(ii) $\phi$

(iii) $\{0,1,2,3,4,5,6,7,8,9,10\}$

(iv) $\{1,2,3,4,5,6,7,8\}$