Question

Given \[A = \left\{ {x|x\,{\rm{is}}\,{\rm{a}}\,{\rm{root}}\,{\rm{of}}\,{x^2} - 1 = 0} \right\}\], \[B = \left\{ {x|x\,{\rm{is}}\,{\rm{a}}\,{\rm{root}}\,{\rm{of}}\,{x^2} - 2x + 1 = 0} \right\}\]. Which of the following is true?
A. \[A \cap B = 1\]
B. \[A \cup B = \phi \]
C. \[A \cup B = A\]
D. \[A \cap B = \phi \]

Answer 1

Hint: First we will find the roster form of the given sets. Then we will find the intersection and union of the given sets. From the union and intersection, we can conclude which statement is true.

Formula Used:
Algebraical identity:
(a) \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]
(b) \[{\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\]

Complete step by step solution:
Given sets are \[A = \left\{ {x|x\,{\rm{is}}\,{\rm{a}}\,{\rm{root}}\,{\rm{of}}\,{x^2} - 1 = 0} \right\}\] and \[B = \left\{ {x|x\,{\rm{is}}\,{\rm{a}}\,{\rm{root}}\,{\rm{of}}\,{x^2} - 2x + 1 = 0} \right\}\].
Now solve the quadratic equation \[{x^2} - 1 = 0\].
\[{x^2} - 1 = 0\]
Now applying the algebraical identity:
 \[\left( {x + 1} \right)\left( {x - 1} \right) = 0\] Since \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]
Equate each factor with zero.
Either,\[x + 1 = 0\]
\[ \Rightarrow x = - 1\]
or,
\[x - 1 = 0\]
\[ \Rightarrow x = 1\]
The roots of \[{x^2} - 1 = 0\] are 1 and -1.
Thus, the roster form of A is \[\left\{ { - 1,1} \right\}\].
Now solve the quadratic equation \[{x^2} - 2x + 1 = 0\].
\[{x^2} - 2x + 1 = 0\]
\[ \Rightarrow {x^2} - 2 \cdot x \cdot 1 + {1^2} = 0\]
Now applying the algebraical identity:
 \[ \Rightarrow {\left( {x - 1} \right)^2} = 0\] Since \[{\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\]
Equate each factor with zero.
Either, \[x - 1 = 0\]
\[ \Rightarrow x = 1\]
or,
\[x - 1 = 0\]
\[ \Rightarrow x = 1\]
The roots of \[{x^2} - 1 = 0\] are 1 and 1.
Thus, the roster form of B is \[\left\{ 1 \right\}\].
Calculate the intersection of set A and set B.
The common element of set A and set B is 1.
So, \[A \cap B = \left\{ 1 \right\}\]
Calculate the union of set A and set B.
\[A \cup B = \left\{ {1, - 1} \right\}\]
             \[ = A\]
Hence option C is the correct option.

Note: Students often do a common mistake to find the intersection of set A and set B. Since the common element of set A and set B is 1. So, they take \[A \cap B = 1\] which is incorrect. But \[A \cap B\] is a set. Thus, the correct form is \[A \cap B = \left\{ 1 \right\}\].