Question

Let F be the set of fractions in simplest form between 0 and 1 that can be written with a single-digit denominator. Find F and |F|​

Answer 1

Answer:

|F| is 27, the whole F is "listed" below.

Step-by-step explanation:

The set of possible denominators is the numbers from 2 to 9. Each of them produces a single fraction for each numerator that's coprime to it.

Let's list those for each:

2: 1

3: 1 2

4: 1 3 (2 is not coprime)

5: 1 2 3 4

6: 1 5 (2 3 and 4 are not coprime)

7: 1 2 3 4 5 6

8: 1 3 5 7

9: 1 2 4 5 7 8 (3 and 6 are not coprime)

To make any number from F, just take the denominator from the rows and the numerator from the list. The size of F is 1 + 2 + 2 + 4 + 2 + 6 + 4 + 6 = 27

Answer 2

The set/number of fractions F for each denominator are ; ( 1, 2, 2, 4, 2, 6, 4, 6 ) .while the value | F | = 27

Given that the set of fractions are between 0 and 10. the possible single digit denominators will be ( 2 through 9 ). also the single digit denominators will have to produce a single fraction where its numerator is coprime to the denominator.

note : coprime numbers are a set of numbers whose HCF (highest common factor ) equals 1

denominators                co-prime numerators

    2                                             1

     3                                           1,  2

     4                                           1,  3  

     5                                           1, 2, 3, 4

     6                                           1, 5  

     7                                           1, 2, 3, 4, 5, 6

     8                                           1, 3, 5, 7

     9                                           1, 2, 4, 5, 7, 8

next : Determine the size of F i.e. | F |

The size of F = ∑ number of coprime numerators for each denominator

                     =  1 + 2 + 2 + 4 + 2 + 6 + 4 + 6 = 27

set of fractions F for each single digit denominator  = ( 1, 2, 2, 4, 2, 6, 4, 6 )

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