NIELIT 2016 MAR Scientist B - Section C: 23

Question

If $S$ be an infinite set and $S​_1​\dots\dots ,S​_n​$ be sets such that $S​_1 ​\cup S​_2​ \cup \dots \cup S​_n​ =S$, then

• A. at least one of the set $S​_i$​ is a finite set.
• B. not more than one of the sets $S​_i​$ can be finite.
• C. at least one of the sets $S​_i​$ is an infinite set.
• D. not more than one of the sets $S​_i​$ can be infinite.

Comments

$\color{red}{\text{Finite Union of Finite Sets is Finite.}}$ i.e. If we have finite number of finite sets, then their union will necessarily be finite.

Union of $S_1,S_2, \dots, S_n$ is infinite set $\text{iff}$ at least one $S_1,S_2, \dots, S_n$ is infinite set. 

Find detailed video solution here: Detailed Video Solution (Click Here) 

Source: https://gateoverflow.in/2301/gate-cse-1993-question-8-3 

Answer 1

$S=S_{1}\bigcup S_{2}\bigcup S_{1}\bigcup S_{3}\bigcup S_{4}\bigcup..........\bigcup S_{n}$

And in question given that S is infinite set

in option A  is wrong  because it say  atleast one of the set $S_{​i}$ is a finite set means if all are finite then S is finite 

in option D not more than one of the sets$S_{​i}$  can be infinite means all are finite same as option  A .  if all are finite then S is finite 

in option B not more than one of the sets $S​_{i}$ can be finite means all $S_{​i}$  are  infinite,if all $S_{​i}$ are infinite so S is infinte so option B is true.

in option C at least one of the sets $S​_{i}$ ​ is an infinite set means if one set is infinite hole S is infinite so option C Is also true

but option C is more correct than option B because atleast one $S​_{i}$ is infinite cover all $S​_{i}$ ​​​​​​​ are infinite

all $S_{​i}$ are infinite $\subset$ atleast one $S​_{i}$ ​​​​​​​ is infinite

so ans is option C :-   S be an infinite set​​​​​​​ if at least one of the sets $S​_{I}$ ​ is an infinite set ​​​​​​​