Question

Make correct statement by filling in the symbol $ \subset $ or $ \not\subset $ in the blank spaces
(i) $\left\{ {2,3,4} \right\}...........\left\{ {1,2,3,4,5} \right\}$
(ii) $\left\{ {a,b,c} \right\}.......\left\{ {b,c,d} \right\}$
(iii) $\left\{ {{\text{X : X is a student of Class XI of your school}}} \right\}.............\left\{ {{\text{X : X student in the school }}} \right\}$
(iv) $\left\{ {{\text{x : x is a circle in the plane }}} \right\}........\left\{ {{\text{x : x is circle in the same plane with radius 1 unit }}} \right\}$
(v) $\left\{ {{\text{x : x is a triangle in a plane}}} \right\}........\left\{ {{\text{x : x is rectangle in the plane}}} \right\}$
(vi) $\left\{ {{\text{x : x is an equilateral triangle in a plane}}} \right\}........\left\{ {{\text{x : x is triangle in the same plane}}} \right\}$
(vii) $\left\{ {{\text{x : x is an even natural number}}} \right\}........\left\{ {{\text{x : x is an integer}}} \right\}$

Answer 1

Hint: As we know that this symbol implies that $ \subset $ subset of given set , If all the element present in the set that is LHS is also present in the element of set that is RHS then we insert the sign $ \subset $ or we can say that it is subset of given set , if it is not the we insert sign $ \not\subset $ .

Complete step-by-step answer:
In this question we have to put $ \subset $ or $ \not\subset $ sign in the blank , this symbol implies that the set is subset or subset not ,
In the part (i) $\left\{ {2,3,4} \right\}...........\left\{ {1,2,3,4,5} \right\}$
As in the LHS the element $2,3,4$ also present in the RHS so we insert $ \subset $ mean it is subset ,
$\left\{ {2,3,4} \right\} \subset \left\{ {1,2,3,4,5} \right\}$
In the part (ii) $\left\{ {a,b,c} \right\}.......\left\{ {b,c,d} \right\}$ , the element in LHS is not present in the RHS ,
Hence $\left\{ {a,b,c} \right\} \not\subset \left\{ {b,c,d} \right\}$
In the part (iii) $\left\{ {{\text{X : X is a student of Class XI of your school}}} \right\}.............\left\{ {{\text{X : X student in the school }}} \right\}$
As we know that those student are present in the Class XI are also present in the school hence it is subset of this ,
$\left\{ {{\text{X : X is a student of Class XI of your school}}} \right\} \subset \left\{ {{\text{X : X student in the school }}} \right\}$
In the part (iv) $\left\{ {{\text{x : x is a circle in the plane }}} \right\}........\left\{ {{\text{x : x is cicle in the same plane with radius 1 unit }}} \right\}$
In this in LHS the radius of circle is not given it could be any radius circle so it doesn't possible that all the point contain in RHS , so
$\left\{ {{\text{x : x is a circle in the plane }}} \right\} \not\subset \left\{ {{\text{x : x is circle in the same plane with radius 1 unit }}} \right\}$
In the part (v)
$\left\{ {{\text{x : x is a triangle in a plane}}} \right\}........\left\{ {{\text{x : x is rectangle in the plane}}} \right\}$
In this triangle and rectangle are different so ,
$\left\{ {{\text{x : x is a triangle in a plane}}} \right\} \not\subset \left\{ {{\text{x : x is rectangle in the plane}}} \right\}$


In the part (vi)
$\left\{ {{\text{x : x is an equilateral triangle in a plane}}} \right\}........\left\{ {{\text{x : x is triangle in the same plane}}} \right\}$
As in this all the equilateral triangle in a plane is the triangle in the same plane or we can say that the equilateral triangle is also the triangle ,hence
$\left\{ {{\text{x : x is an equilateral triangle in a plane}}} \right\} \subset \left\{ {{\text{x : x is triangle in the same plane}}} \right\}$
In the part (vii)
$\left\{ {{\text{x : x is an even natural number}}} \right\}........\left\{ {{\text{x : x is an integer}}} \right\}$
So the even natural number is $\left\{ {2,4,6,8..........} \right\}$ and in the integer is $\left\{ { - \infty ....... - 1,0,1..........\infty } \right\}$ so all the element in LHS is in RHS hence
$\left\{ {{\text{x : x is an even natural number}}} \right\} \subset \left\{ {{\text{x : x is an integer}}} \right\}$

Note: Power Set :
In set theory, the power set (or power set) of a Set A is defined as the set of all subsets of the Set A including the Set itself and the null or empty set. It is denoted by P(A). Basically, this set is the combination of all subsets including null set, of a given set.