Download CPSC 121 - MATHEMATICAL PROOFS 1. Proofs in General

CPSC 121 - MATHEMATICAL PROOFS
1. Proofs in General
Problem 1. Determine the truth value of each of these statements.
(1) ∃n ∈ Z, ∃m ∈ Z, n2 + m2 = 5
(2) ∃n ∈ Z, ∃m ∈ Z, n + m = 4 ∧ n − m = 2
(3) ∀n ∈ Z, n < n2
(4) ∀n ∈ Z, (n − 5)2 < (n + 5)2
(5) ∃n ∈ Z, ∃m ∈ Z, n2 + m2 = 6
(6) ∃n ∈ Z, ∃m ∈ Z, n + m = 4 ∧ n − m = 1
(7) ∀n ∈ Z, ∃m ∈ Z, n2 < m
(8) ∀n ∈ Z, ∃m ∈ Z, n + m = 0
(9) ∃n ∈ Z, ∀m ∈ Z, n < m2
(10) ∃n ∈ Z, ∀m ∈ Z, nm = m
(11) ∀n ∈ Z, ∀m ∈ Z, ∃p ∈ Z, p = m+n
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Problem 2.
Prove ∀x ∈ R, x > 1 → x2 > x.
Problem 3. Prove that for every distinct pair of real numbers, there is another real number that is between them
(greater than the smaller one and less than the larger one).
Problem 4. Prove that the fourth power of a positive odd integer can be written in the form 8m + 1, where m is
a non-negative integer. Hint: a positive odd integer can be written as 2i + 1, where i is a non-negative integer.
Definition. The floor function assigns to the real number x the largest integer that is less than or equal
to x. In other words, the floor function rounds a real number down to the nearest integer. The value of
the floor function is denoted by bxc.
Definition. The ceiling function assigns to the real number x the smallest integer that is greater than
or equal to x. In other words, the ceiling function rounds a real number up to the nearest integer. The
value of the ceiling function is denoted by dxe.
√
√
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Example. bπc = 3. dπe = 4. b− c = −1. d− e = 0. b 2c = 1. d 2e = 2. b−5c = −5. d−5e = −5.
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Problem 5. Prove that for any positive integer x, if x is one more than a multiple of 3, then the sum of the floor
of x3 , the floor of x3 and, the ceiling of x3 is equal to x.
Problem 6. For any integer m, if m is a perfect square, then m + 2 is not a perfect square.
2. Proof by Contraposition
Problem 7. Prove that for any integer n, if n2 + 8n − 1 is even, then n is odd.
Problem 8. Prove that for any integers a, b and n, if n - (a · b), then n - a and n - b.
Problem 9. Prove that for any x, y ∈ R, if y 3 + yx2 ≤ x3 + xy 2 , then y ≤ x.
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CPSC 121 - MATHEMATICAL PROOFS
3. Proof by Contradiction
Problem 10. For any two real numbers a and b, if a is rational and ab is irrational, then b is irrational.
Problem 11. There do not exist integers a and b such that 7a + 21b = 1.
Problem 12. For any integers a, b, and c, if a2 + b2 = c2 , then a is even or b is even.
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