Download CPSC 121 - PROOFS Problem 1. Determine the truth value of each

CPSC 121 - PROOFS
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 up 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 down 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.
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