Question

For any two sets A and B, \[A=B\] is equivalent to
(A) \[A-B=B-A\]
(B) \[A\cup B=A\cap B\]
(C) \[A\cup C=B\cup C\] and \[A\cap C=B\cap C\] for any set C
(D) \[A\cap B=\phi \]

Answer 1

Hint: It is given that \[A=B\] . First of all, check option (A). The LHS is equal to \[\left( A-B \right)\] . Now, replace A by B and B by A in the equation \[\left( A-B \right)\] and check whether the LHS is equal to RHS or not. Now, check option (B). The LHS and RHS is equal to \[\left( A\cup B \right)\] and \[\left( A\cap B \right)\] respectively. Now, replace A by B in the LHS and RHS. We know the property that the intersection and union of a set with the same set is the set itself. Now, use this property and check whether the LHS and RHS hold the same result or not. Then, check option (C). It has two parts. The first part is \[A\cup C=B\cup C\] and the second part is \[A\cap C=B\cap C\] . Replace A by B in the LHS of the first part and check whether it is equal to RHS or not. Similarly, replace A by B in the LHS of the second part and check whether it is equal to RHS or not. Now, check option (D). In LHS, we have \[A\cap B\] and in RHS we have, \[\phi \] . Replace A by B in the LHS. We know the property that the intersection of a set with the same set is the set itself. Now, check whether the LHS and RHS hold the same result or not.

Complete step-by-step answer:
According to the question, we have two sets A and B,
\[A=B\] ……………………..(1)
We have four options A, B, C, and D,
In option (A), we have \[\left( A-B \right)\] as LHS and \[\left( B-A \right)\] as RHS.
 LHS= \[\left( A-B \right)\] ………………..(2)
RHS = \[\left( B-A \right)\] ……………………(3)
From equation (1), we have \[A=B\] .
Now, replacing A by B and B by A in equation (2), we get
LHS = \[A-B=B-A\] ……………………..(4)
From equation (3), we have RHS equal to \[\left( B-A \right)\] .
So, LHS=RHS.
Hence option (A) is the correct one.
In option (B), we have \[\left( A\cup B \right)\] as LHS and \[\left( A\cap B \right)\] as RHS.
 LHS= \[\left( A\cup B \right)\] ………………..(5)
RHS = \[\left( A\cap B \right)\] ……………………(6)
From equation (1), we have \[A=B\] .
Now, replacing A by B in equation (5) and equation (6), we get
LHS = \[\left( A\cup B \right)=\left( B\cup B \right)\] ……………………..(7)
RHS = \[\left( A\cap B \right)=\left( B\cap B \right)\] …………………………..(8)
We know the property that the intersection and union of a set with the same set is the set itself.
Using this property in equation (7) and equation (8), we get
LHS = \[\left( B\cup B \right)=B\] ………………………..(9)
RHS = \[\left( B\cap B \right)=B\] ……………………………(10)
So, LHS=RHS.
Hence option (B) is the correct one.
In option (C), we have two parts. One is \[A\cup C=B\cup C\] and the other part is \[A\cap C=B\cap C\] .
In the first part, we have \[A\cup C=B\cup C\] .
LHS = \[\left( A\cup C \right)\] …………………………..(11)
RHS = \[\left( B\cup C \right)\] ……………………………(12)
From equation (1), we have \[A=B\] .
Now, replacing A by B in equation (11), we get
LHS = \[\left( A\cup C \right)=\left( B\cup C \right)\] ……………………(13)
From equation (12), we have RHS equal to \[\left( B\cup C \right)\] .
So, LHS = RHS.
Therefore, the first part is correct.
In the second part, we have \[A\cap C=B\cap C\] .
LHS = \[\left( A\cap C \right)\] …………………………..(14)
RHS = \[\left( B\cap C \right)\] ……………………………(15)
From equation (1), we have \[A=B\] .
Now, replacing A by B in equation (14), we get
LHS = \[\left( A\cap C \right)=\left( B\cap C \right)\] ……………………(16)
From equation (15), we have RHS equal to \[\left( B\cap C \right)\] .
So, LHS = RHS.
Therefore, the second part is also correct.
Hence, the option is also correct.
In option (D), we have \[A\cap B\] as LHS and \[\phi \] as RHS.
LHS = \[\left( A\cap B \right)\] ………………………………..(17)
RHS = \[\phi \] …………………………..(18)
From equation (1), we have \[A=B\] .
Now, replacing A by B in equation (17), we get
LHS = \[\left( A\cap B \right)=\left( A\cap A \right)\] ……………………………(19)
We know the property that the intersection of a set with the same set is the set itself.
Using this property in equation (19), we get
LHS = \[\left( A\cap A \right)=A\] ………………………………(20)
From equation (18), we have the RHS equal to \[\phi \] .
So, \[LHS\ne RHS\] .
Therefore, option (D) is not correct.
Hence, the correct options are (A), (B), and (C).

Note: The best way to solve this type of question is to check each and every option given. If the LHS and RHS of an option holds the same result then that option is correct. Here, one might make a mistake and take the union and intersection of the same set as \[\phi \] . This is wrong because the union and intersection of the same set is the set itself.