Question

Determine whether the statement “If $J = \left\{ {15,24,33,42,51,60} \right\}$, the set builder form of J is $J = \left\{ {b|{\rm{\text b\ is\ a\ two\ digit\ number\ having\ sum\ of\ digits\ is\ 6}}} \right\}$” is true or false.

Answer 1

Hint: Start by adding each digit of the number in the set J. For all the 6 numbers inside the J check whether the sum is 6 or not. If the sum is equal to 6, it can be concluded that the given set builder form represents the set.

Complete step-by-step answer:
Let the two-digit number b is of the form $b = xy$. This implies that 15 can be written as $ x = 1,y = 5$.
Check whether the sum of two-digit number 15 is 6:
$\Rightarrow$ $1 + 5 = 6$
Check whether the sum of two-digit number 24 is 6:
$\Rightarrow$$2 + 4 = 6$
Check whether the sum of two-digit number 33 is 6:
$\Rightarrow$$3 + 3 = 6$
Check whether the sum of two-digit number 42 is 6:
$\Rightarrow$$4 + 2 = 6$
Check whether the sum of two-digit number 51 is 6:
$\Rightarrow$$5 + 1 = 6$
Check whether the sum of two-digit number 60 is 6:
$\Rightarrow$$6 + 0 = 6$
Therefore, from the above calculations it can be concluded that the set builder form of J is $J = \left\{ {b|{\rm{\text b\ is\ a\ two\ digit\ number\ having\ sum\ of\ digits\ is\ 6}}} \right\}$”
Hence, the statement is true.

Note:
There is a chance of making an error while taking a sum of two numbers and also for each number the result that the sum is 6 has to be verified. Only when all the numbers in set J have their sum of digits as 6, the set builder form will be valid.