Question

Are the following pair of sets equal ? Give reasons .
(i) A = \left\\{ {2,3} \right\\} , B = \left\\{ {{\text{x : x is a solution of }}{{\text{x}}^2} + 5{\text{x + 6 = 0}}} \right\\}
(ii) A = \left\\{ {{\text{x : x is a letter in the word FOLLOW}}} \right\\} , B = \left\\{ {{\text{y : y is a letter in the word WOLF}}} \right\\}

Answer 1

So we know that the equal set are those sets in which the element present in the set are same or equal , In part (i) Set A is given for the Set B ,solve the equation ${{\text{x}}^2} + 5{\text{x + 6 = 0}}$ and value of x is the set B , In part (ii) Set A is letter in FOLLOW and in Set B is the letter in WOLF .

Complete step-by-step answer:
As in this question we have to find out whether both sets are equal or not , from the definition of equal set that Equal sets have the exact same elements in them, even though they could be out of order.
For the Part (i) that is A = \left\\{ {2,3} \right\\} , B = \left\\{ {{\text{x : x is a solution of }}{{\text{x}}^2} + 5{\text{x + 6 = 0}}} \right\\}
So it is given in the question that Set A = \left\\{ {2,3} \right\\} ,
Now for the set B = \left\\{ {{\text{x : x is a solution of }}{{\text{x}}^2} + 5{\text{x + 6 = 0}}} \right\\}
In the Set B ${\text{x : x is a solution of }}{{\text{x}}^2} + 5{\text{x + 6 = 0}}$
we have to find the solution of ${{\text{x}}^2} + 5{\text{x + 6 = 0}}$ i.e ,
${x^2} + 2x + 3x + 6$ = $0$
Now on solving this we get $(x + 2)(x + 3)$ , and $x = - 2, - 3$
Hence the Set B contains B = \left\\{ { - 2, - 3} \right\\}
Set A = \left\\{ {2,3} \right\\} Set B = \left\\{ { - 2, - 3} \right\\}
Hence both are not equal because both does not contains same element ,
Set A $\ne$ Set B
Now for the Part (ii) A = \left\\{ {{\text{x : x is a letter in the word FOLLOW}}} \right\\} , B = \left\\{ {{\text{y : y is a letter in the word WOLF}}} \right\\}
In the Set A the element is the letter that are present in the FOLLOW ,
So from this we can say that the set A Contains \left\\{ {{\text{F,O,L,W}}} \right\\}
In the Set B it is given that it contains the letter in the word WOLF ,
So from this we can say that the set B = \left\\{ {{\text{W,O,L,F}}} \right\\}
So from this Set A = \left\\{ {{\text{F,O,L,W}}} \right\\} and Set B = \left\\{ {{\text{W,O,L,F}}} \right\\}
Hence the element in both the set is equal therefore ,
Set A = Set B

Note: Empty Relation
An empty relation (or void relation) is one in which there is no relation between any elements of a set , hence it is an empty relation or void relation.