Question

State whether the following statement is true or false. Give reason to support your answer.
\[\left\{ {a,b,c} \right\}\,\,and\,\,\left\{ {1,2,3} \right\}\] are equivalent sets.
 $
  A.\,\,True \\
  B.\,\,False \\
  $

Answer 1

Hint: To discuss whether two or more sets are equal or equivalent sets. We first see how many elements they have and then we see whether given sets are having the same elements or different irrespective of their position. Then according to this we can name given sets either equal or equivalent sets.

Complete step-by-step answer:
Given two sets \[\left\{ {a,b,c} \right\}\,\,and\,\,\left\{ {1,2,3} \right\}\]
To discuss whether given sets are equivalent sets or not.
For this we first discuss definitions of both equal and equivalent sets.
Equal sets: Two or more sets are said to be equal if they are having the same number of the same elements.
Means number of elements in given sets are equal moreover elements are also same.
For example let consider following sets:
 $ A\left\{ {a,b,c} \right\},\,\,B\left\{ {c,b,a} \right\}\,\,and\,\,C\left\{ {b,a,c} \right\} $
Here, we have three sets A, B and C.
All these three sets having number of elements = $ 3 $
More ever, elements in all three sets are also the same.
Hence, we can say that three sets A, B and C are equal sets.
Now, we discuss equivalent sets.
Equivalent sets are those sets in which the number of elements are the same but the elements are different.
So, we can say that two more sets are said to be equivalent if they are having the same number of different elements.
For example let consider following sets:
 $ A\left\{ {1,2,3} \right\}\,\,and\,\,B\left\{ {a,b,c} \right\} $
Clearly in the above sets we see that both sets A and B are having the same number of elements. Which are equal to $ 3 $ but we see that elements in both sets are different.
In set elements are numeric where elements in set B are alphabets.
But their number of elements in both sets are the same.
Hence, from above we can say that both sets A and B are equivalent as they both have the same number of different elements.
Therefore, from above discussion we can say that given sets \[\left\{ {a,b,c} \right\}\,\,and\,\,\left\{ {1,2,3} \right\}\] are also said to be \[\left\{ {a,b,c} \right\}\,\,and\,\,\left\{ {1,2,3} \right\}\] but not equal as both sets have same number of different elements.
So, the correct answer is “TRUE”.

Note: Two sets are said to be equivalent if they are having only the same cardinal number but their elements can be different where in two or more equal sets their cardinal numbers as well as elements must be the same. So using this concept we can discuss whether given sets are equal or equivalent sets.