Question

Let \[X\] and \[Y\] be two non-empty sets such that \[X\cap A=Y\cap A=\varnothing \] and \[X\cup A=Y\cup A\] for some non-empty set $A$. Then,
A. \[X\] is a proper subset of \[Y\]
B. \[Y\] is a proper subset of \[X\]
C. \[X=Y\]
D. \[X\] and \[Y\] are disjoint sets

Answer 1

Hint: In this question, we are to find the relation between the given non-empty sets \[X\] and \[Y\]. The given conditions are used to extract the relation between \[X\] and \[Y\].



Formula Used:Set: A collection of objects in which it is possible to decide whether a given object belongs to the collection or not is said to be a set. Those objects are nothing but the elements in the set.
Sets are represented in two ways: The roaster method and the Set builder form
Roaster method:
The set of vowels – \[\left\{ a,\text{ }e,\text{ }i,\text{ }o,\text{ }u \right\}\]
The set of natural numbers – \[\left\{ 1,\text{ }2,\text{ }3,\text{ }\ldots . \right\}\]
Set builder form:
The set of vowels – \[\left\{ x:x\text{ }is\text{ }a\text{ }vowel\text{ }in\text{ }English\text{ }alphabet \right\}\]
The set of natural numbers – \[\left\{ x:x\text{ }is\text{ }a\text{ }natural\text{ }number \right\}\]
Some of the important mathematical sets:
$N$ - the set of natural numbers - $N=\{1,2,3...\}$
$Z$- the set of integers - $Z=\{0,\pm 1,\pm 2,\pm 3,...\}$
If two sets $A, B$ where the set $A$ is said to be the subset of $B$ i.e., $A\subseteq B$ then every element of $A$ is in the set $B$ and the set $A$ is said to be the proper subset of $B$ i.e., \[A\subset B\] then $A\subseteq B$ and $A\ne B$.
If two sets $A, B$ are given by the relation $A=B$, then they contain the same elements.



Complete step by step solution:It is given that there are three non-empty sets as $X, Y, A$
And they are related by
\[X\cap A=Y\cap A=\varnothing \]
\[X\cup A=Y\cup A\]
Consider an element $a$.
Since we have \[X\cap A=\varnothing \], we can write \[a\in X,a\notin A\] or \[a\in A,a\notin X\]
Similarly, we have \[Y\cap A=\varnothing \], then we can write \[a\in Y;a\notin A\] or \[a\in A;a\notin Y\]
So, from these points, we can say that \[a\in X,Y\text{ }...(1)\]
We have also given that \[X\cup A=Y\cup A\]. So, we can write
\[a\in X\cup A;a\in Y\cup A\text{ }...(2)\]
Thus, from (1) and (2),
\[X=Y\]



Option ‘C’ is correct
Note: Here we need to assume any element and apply the given conditions for that element. Then we can able to extract the required relationship between the given sets.