COMP232 - Mathematics for Computer Science
Tutorial 6
Ali Moallemi
moa [email protected]
Iraj Hedayati
h [email protected]
Concordia University, Winter 2016
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
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Table of Contents
1
1.8 Rules of Inference
Exercise 8
Exercise 14
Exercise 19
Exercise 31
Exercise 38
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
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Exercise 8
Prove that there is a positive integer that equals the sum of the positive
integers, not exceeding it. Is your proof constructive or non constructive.
Answer: 3 = 1 + 2
Constructive.
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
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Exercise 14
Prove or disprove that, if a and b are rational numbers, then ab is also
rational.
Answer:
To disprove
it we just need to find a counter example:
√
1
2
2 = 2
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
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Exercise 19
Show that if n is an odd integer, then there is a unique integer k such that
n is the sum of k − 2 and k + 3.
Answer: Let n be any arbitrary odd integer. We look for a k such that,
n = (k − 2) + (k + 3) that is, n = 2k + 1. To find k, it is sufficient to see
n−1
that k = n−1
2 . Since n is odd then, n − 1 is even, consequently, 2 is an
integer, which means k is an integer. Moreover, this equation has only one
solution that is, k is unique. Finally, There is a unique integer k for any n.
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
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Exercise 31
Prove that there are no solutions in positive integers x and y to the
equation x4 + y 4 = 625.
Answer: On one hand, since 54 = 625, both x and y have to be less than
5. On the other hand if both x and y be 4, Then we have,
x4 + y 4 = 44 + 44 = 512, and 512 < 625. Then there is no integer value
for x and y such that x4 + y 4 = 625.
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
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Exercise 38
Prove or disprove that if you have an 8-gallon jug of water and two empty
jugs with capacities of 5 gallons and 3 gallons, respectively, then you can
measure 4 gallons by successively pouring some of or all of the water in a
jug into another jug.
Answer: Let (j1 , j2 , j3 ) represent the quantities of water in each jug. Here
is the sequence of steps.
1- (8, 0, 0)
2- (3, 5, 0)
3- (3, 2, 3)
4- (6, 2, 0)
5- (6, 0, 2)
6- (1, 5, 2)
7- (1, 4, 3)
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
7/7