Discrete Computational Exercises Section 2-5 Essay

COT3103
10/21/2014
Exercises Section 2.5

2. Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set.
a. The integers greater than 10.
This is countably infinite.
Starting from the first integer greater than 10, which is 11, one can infinitely count upwards since there is no boundary on the right side of the number line for this instance.
The equation ƒ(x) = x + 11 can be used to show a one-to-one correspondence. x: 1, 2, 3, 4, 5, 6, …

ƒ(x): 11, 12, 13, 14, 15, 16 …
b. The odd negative integers.
This is countably infinite.
Starting from the first odd negative integer closest to 0, which …show more content…

a. Integers not divisible by 3.
This set is countable.
The integers in the set are ±1, ±2, ±4, ±5, ±7, and so on.
We can list these numbers in the order 1, −1, 2, −2, 4, −4, 5, −5, 7, −7, ... , thereby establishing the desired correspondence. x: 1, 2, 3, 4, 5, 6, …

ƒ(x): 1, -1, 2, -2, 3, -3, …
b. Integers divisible by 5 but not by 7.
This is countably infinite.
This is similar to part (a); we can simply list the elements of the set in order of increasing absolute value, listing each positive term before its corresponding negative: (x): 1, 2, 3, 4, 5, 6, 7, 8, 9…

ƒ(x): 5, -5, 10, -10, 15, -15, 20, -20, 25, -25, 30, -30 40, -40, …
c. The real numbers with decimal representations consisting of all 1s.
This set is countable but a little tricky.
We can arrange the numbers in a 2-dimensional table as follows:

Thus we have shown that our set is the countable union of countable sets (each of the countable sets is one row of this table).
Therefore by Exercise 27, the entire set is countable.
For an explicit correspondence with the positive integers, we can zigzag along the positive-sloping diagonals as in Figure 3: 1↔ .1 , 2↔ 1.1 , 3↔.1, 4↔.11, 5↔1, and so on.
d. The real numbers with decimal representations of all 1s or 9s.
This set is not countable.
We can prove it by the same diagonalization argument as was used to prove that the set of all reals is uncountable in Example 5.
All we need to do is choose di =