Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

Selected Solutions 339
(b) We get the same set as we did in the previous part, and the smallest
non-negative number for which n 2 − 5 is a natural numbers is 3.
Note that if we didn’t specify n ∈ N then any integer less than −3
would also be in the set, so there would not be a least element.
(c) This is the set {1, 2, 5, 10, . . .}, namely the set of numbers that are
the result of squaring and adding 1 to a natural number. (0 2 + 1 1,
1 2 + 1 2, 2 2 + 1 5 and so on.) Thus the least element of the set is 1.
(d) Now we are looking for natural numbers that are equal to taking
some natural number, squaring it and adding 1. That is,
{1, 2, 5, 10, . . .}, the same set as the previous part. So again, the least
element is 1.
0.3.3.
(a) 34. Note that 37 − 4 33, but this calculation would not include 4
itself.
(b) 103. Again, you could compute this by 100 − (−2) + 1, or simply
count: 100 numbers from 1 through 100, plus -2, -1, and 0.
(c) 8. There are 8 primes not greater than 20: {2, 3, 5, 7, 11, 13, 17, 19}.
0.3.4. {2, 4}.
0.3.5. {1, 2, 3, 4, 5, 6, 8, 10}
0.3.6. 11.
0.3.7. There will be exactly 4 such sets: {2, 3, 4}, {1, 2, 3, 4}, {2, 3, 4, 5}
and {1, 2, 3, 4, 5}.
0.3.8.
(a) A ∩ B {3, 4, 5}.
(b) A ∪ B {1, 2, 3, 4, 5, 6, 7}.
(c) A \ B {1, 2}.
(d) A ∩ (B ∪ C) {1}.
0.3.9.
(a) A ∩ B will be the set of natural numbers that are both at least 4
and less than 12, and even. That is, A ∩ B {x ∈ N : 4 ≤ x <
12 ∧ x is even} {4, 6, 8, 10}.
(b) A \ B is the set of all elements that are in A but not B. So this is
{x ∈ N : 4 ≤ x < 12 ∧ x is odd} {5, 7, 9, 11}.
Note this is the same set as A ∩ B.
0.3.11. For example, A {2, 3, 5, 7, 8} and B {3, 5}.