Question

Show that $A \cup B = A \cap B$ implies that $A = B$.

Answer 1

Here we will first assume an element of the set A and then we will form the condition between set A and set B in terms of the subset. Then we will assume an element belongs to set B and form the condition between set A and set B in terms of the subset. Then by comparing these conditions we will get the required expression.

Complete step-by-step answer:
Let $x$ be an element which belongs to set A i.e. $x \in A$.
It means that $x$ will also belong to the union of set A and set B i.e. $x \in A \cup B$.
It is given that $A \cup B = A \cap B$ which means $x$ will also belong to the intersection of set A and set B i.e. $x \in A \cap B$. By this, we can say that the element belongs to the set B also i.e. $x \in B$.
Therefore, by this, we can say that if the element belongs to set A, then it must belong to the set B which means set A is the subset of set B.
$\Rightarrow A\subset B$……………………..$\left( 1 \right)$
Similarly, let $y$ be an element which belongs to set B or $y \in B$.
Therefore, it means that $y$ will also belong to the union of set A and set B i.e. $y \in A \cup B$.
It is given that $A \cup B = A \cap B$ which means $y$ will also belong to the intersection of set A and set B i.e. $y \in A \cap B$. By this, we can say that the element belongs to set A also i.e. $y \in A$.
Therefore, by this, we can say that if the element belongs to set B, then it must belong to set A which means set B is the subset of set A.
$\Rightarrow B\subset A$……………………..$\left( 2 \right)$
From equation $\left( 1 \right)$ and equation $\left( 2 \right)$, we can say that set A equals the set B.
$\Rightarrow A=B$
Hence proved.

Note: Here we have to note that the set which includes all the elements from every set of data is called as union and denoted as $A \cup B$. The set which includes only common terms between the given sets is called an intersection and generally denoted as $A \cap B$. A subset is the set of the elements whose elements are present in the other main set. It is denoted as $A \subset B$. If the two sets are given and those two sets are the subsets to each other, then both the sets are equal.