Question

If \[A =\{ x:{x^2}=1\} \] and \[B=\{ x:{x^4}=1\} \] then \[A\Delta B\] is equal to \[x:C\], where \[C\] is
A. \[\{ - 1,1,i, - i\} \]
B. \[\{ - 1,1\} \]
C. \[\{ i, - i\} \]
D. \[\{ - 1,1,i\} \]

Answer 1

Hint: First the elements of the two sets \[A\] and \[B\] are found out by solving the given equations in each set and then using the formulae for symmetric difference of two sets, the elements of \[A\Delta B\] are found out to decide the correct option.
Formulae Used: If \[A\]and \[B\]are any two sets, then \[A\Delta B = (A - B) \cup (B - A)\]
Where, \[A - B = \{ x:x \in A\]and \[ \notin B\} \]
              \[B - A = \{ x:x \in B\]and \[ \notin A\} \]
And \[(A - B) \cup (B - A) = \{ x:x \in (A - B)\] or \[(B - A)\} \]

Complete step-by-step solution:
We have been given the set \[A\] as \[A = \{ x:{x^2} = 1\} \]
Let us find out the elements of \[A\]by solving the equation \[{x^2} = 1\]
\[{x^2} = 1\]
\[ \Rightarrow x = \pm 1\]
So, \[A = \{ + 1, - 1\} \]
Similarly we will find out the elements of \[B\] by solving the equation \[{x^4} = 1\]
\[{x^4} = 1\]
\[ \Rightarrow {x^2} = \pm 1\]
So, \[{x^2} = + 1\] or \[{x^2} = - 1\]
Considering the first case
\[{x^2} = + 1\]
\[ \Rightarrow x = \pm 1\]
Considering the second case
\[{x^2} = - 1\]
\[ \Rightarrow x = \pm i\]
So, \[B = \{ + 1, - 1, + i, - i\} \]
We know that \[A - B = \{ x:x \in A\]and \[ \notin B\} \]
So, \[A - B = \{ + 1, - 1\} - \{ + 1, - 1, + i, - 1\} = \emptyset \], which is a null set and has no element.
Similarly, \[B - A = \{ x:x \in B\]and \[ \notin A\} \]
So, \[B - A = \{ + 1, - 1, + i, - i\} - \{ + 1, - 1\} = \{ + i, - i\} \]
Let, the symmetric difference between the two sets \[A\] and \[B\] is the set \[C\]
So, \[A\Delta B = C = \{ x:x \in C\} \]
Now, we will apply the formulae to find \[A\Delta B\]
We know \[A\Delta B = (A - B) \cup (B - A)\]
              \[ \Rightarrow C = \emptyset \cup \{ + i, - i\} \] (Since \[A\Delta B = C,(A - B) = \emptyset ,(B - A) = \{ + i, - i\} \])
Now, applying the union rule i.e. \[A \cup B = \{ x:x \in A\]or \[x \in B\} \], we have
\[A\Delta B = \emptyset \cup ( + i, - i\} = \{ + i, - i\} \]
Hence, \[A\Delta B = C = \{ i, - i\} \]

Therefore, option C. \[\{ i, - i\} \] is the correct answer.

Note: The symmetric difference between two sets \[A\]and \[B\] denoted by the symbol \[A\Delta B\] can be found by the union of two sets, which are derived by finding relative complements of the two sets \[A\]and \[B\] denoted by the symbol \[(A - B) \cup (B - A)\]. Intersection i.e. \[(A - B) \cap (B - A)\]should not be used to find the required symmetric difference.