Question

A and B are two sets such that \[A\subset B\] then what is the value of \[A\cup B\]?

Answer 1

Hint: We solve this problem by using the formula of the union of sets.
We have the formula of the union of sets given as
\[A\cup B=A+B-A\cap B\]
We find the value of \[A\cap B\] that is the set of common elements in both A and B by using the condition that A is the subset of B
We have the definition of a subset that if every element in set \[x\] also present in set \[y\] then \[x\] is said to be subset o \[y\] and it is denoted as \[x\subset y\]

Complete step by step answer:
We are given that there are two sets A and B
We are also given that \[A\subset B\]
We know that the definition of the subset that if every element in set \[x\] also present in set \[y\] then \[x\] is said to be subset o \[y\] and it is denoted as \[x\subset y\]
By using the above definition we can say that A is a subset of B and every element of A is also present in B
We know that the intersection of sets is given as the common elements in the two sets.
Here we can see that the common elements in A and B are elements of A because all elements of A are present in B
By using the above condition we get the intersection of A and B as
\[\Rightarrow A\cap B=A\]
Now, let us find the union of sets.
We know that the formula of union of sets given as
\[A\cup B=A+B-A\cap B\]

By using the above formula we get
\[\begin{align}
  & \Rightarrow A\cup B=A+B-A \\
 & \Rightarrow A\cup B=B \\
\end{align}\]
Therefore we can conclude that the union of given two sets is B that is
\[\therefore A\cup B=B\]


Note:
 We can solve this problem by using the Venn diagrams
We are given that A and B are two sets such that \[A\subset B\]
We know that the definition of subset that if every element in set \[x\] also present in set \[y\] then \[x\] is said to be subset o \[y\] and it is denoted as \[x\subset y\]
By using the definition of subset for the given sets we get the Venn diagram as follows

Here, we can see that the shaded region is set B.
We know that the union of sets is a set containing all the elements of two sets along with common elements.
We know that the elements in a set repeat only once.
By using the above two conditions and the Venn diagram, we get that the union of sets is B because set B includes all elements of A and B along with common elements.
Therefore we can conclude that the union of given two sets is B that is
\[\therefore A\cup B=B\]