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

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.