Discrete Computational Exercises Section 2-5
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 is -1, we can count backwards infinitely without a boundary on the left side of the number line for this instance.
The equation ƒ(x) = -2x + 1 can be used to show a one-to-one correspondence. x: 1, 2, 3, 4, 5, 6, …
ƒ(x): -1, -3, -5, -7, -9, -11, …
c. The integers with absolute value less than 1,000,000.
Finite.
This set is {−999,999, −999,998, ... ,−1, 0, 1, ... , 999,999}.
It is finite, with cardinality 1,999,999.
d. The real numbers between 0 and 2.
Uncountable.
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.
e. The set A × Z+ where A = {2,3}.
This set is countable.
We can list its elements in the order (2,1), (3,1), (2,2), (3,2), (2,3), (3,3), ... giving us the one-to-one correspondence 1↔(2,1), 2↔(3,1), 3↔(2,2), 4↔(3,2), 5↔(2,3), 6↔(3,3), ...
x: 1, 2, 3, 4, 5, 6, …
ƒ(x): (2,1), (3,1), (2,2), (3,2), (2,3), (3,3) …
f. The integers that are multiples of 10.
This 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): 0, 10, -10, 20, -20, 30, …
4. Determine whether each of
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 is -1, we can count backwards infinitely without a boundary on the left side of the number line for this instance.
The equation ƒ(x) = -2x + 1 can be used to show a one-to-one correspondence. x: 1, 2, 3, 4, 5, 6, …
ƒ(x): -1, -3, -5, -7, -9, -11, …
c. The integers with absolute value less than 1,000,000.
Finite.
This set is {−999,999, −999,998, ... ,−1, 0, 1, ... , 999,999}.
It is finite, with cardinality 1,999,999.
d. The real numbers between 0 and 2.
Uncountable.
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.
e. The set A × Z+ where A = {2,3}.
This set is countable.
We can list its elements in the order (2,1), (3,1), (2,2), (3,2), (2,3), (3,3), ... giving us the one-to-one correspondence 1↔(2,1), 2↔(3,1), 3↔(2,2), 4↔(3,2), 5↔(2,3), 6↔(3,3), ...
x: 1, 2, 3, 4, 5, 6, …
ƒ(x): (2,1), (3,1), (2,2), (3,2), (2,3), (3,3) …
f. The integers that are multiples of 10.
This 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): 0, 10, -10, 20, -20, 30, …
4. Determine whether each of