Question

Show that \[A \cup B = A \cap B\] implies that \[A = B\].

Answer 1

Hint: 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.