(a) Illustrate a one-to-one correspondence between the following sets:
N = {1, 2, 3, 4,. . . , n,. . .}
F = {4, 9, 14, 19, 24,. . .}
(b) In your correspondence, what element of F corresponds to 57? Explain why.
(c) In your correspondence, what element of F corresponds to n?
I see a mapping
such that n ----> 5n - 1
How is that for a hint?
(a) To illustrate a one-to-one correspondence between the sets N and F, we need to pair each element of N with a unique element of F in such a way that no two elements of N are paired with the same element of F, and vice versa.
One way to do this in this case is to define a correspondence where each element of N is paired with an element of F using the formula: f = 5n - 1.
So, the correspondence between N and F would look like this:
1 → 4
2 → 9
3 → 14
4 → 19
...
n → 5n - 1
...
(b) To find the element of F that corresponds to 57, we can use the formula we established in part (a): f = 5n - 1. We can solve this equation for n to determine the corresponding element from N.
For 57, we have:
57 = 5n - 1
Adding 1 to both sides:
58 = 5n
Dividing both sides by 5:
n = 58/5 = 11.6
Since n must be an integer, we round this down to the nearest whole number: n = 11.
Therefore, the element of F that corresponds to 57 is found by substituting n = 11 into the formula: f = 5n - 1.
f = 5(11) - 1 = 54.
(c) In the correspondence between N and F, the element of F that corresponds to n can be determined using the same formula we established: f = 5n - 1.
So, the element of F that corresponds to n is given by substituting the value of n into the formula: f = 5n - 1.