Need help with Discrete Mathematics with Applications, have something that is due tonight at 12 Midnight.

 

 

CISC502 Assignment 1
Exercise Set 1.1 (p. 5+): Exercise 2, 4
2. Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when
it is divided by 6?
a) Is there an integer n such that n has _______?
b) Does there exist _____ such that if n is divided by 5 the remainder is 2 and if _____?
Let n = 3j + 2, where j is a positive integer
Let t = 5k + 3, where k is a positive integer
nt = (3j+2)(5k+3) = 15jk + 9j + 10k + 6
So the question is: what is the remainder after 15jk + 9j + 10k + 6 is divided by 15? Well, 15jk is
clearly divisible by 15. If we show that 9j and 10k are divisible by 15 as well, then we can
determine the remainder (6).
(1) The fact that n - 2 is divisible by 5 means that 3j is divisible by 5. So j can be written as 5x,
where x is a positive integer. That means we can rewrite 9j (in the question) as 45x, which is
divisible by 15. But we don't know whether 10k is divisible by 15. Insufficient.
(2) That it is divisible by 3 means that 5k + 3 is divisible by 3, and therefore 5k is divisible by 3.
So k can be written as 3y, where y is a positive integer. That means we can rewrite 10k (in the
question) as 30y, which is divisible by 15. But we don't know whether 9j is divisible by 15.
Insufficient.
(1&2) nt = 15jk + 9j + 10k + 6 = 15jk + 45x + 30y + 6. The remainder must be 6. Sufficient.
4. Given any real number, there is a real number that is greater.
a) Given any real number r, there is ____ s such that s is _____.
b) For any ___, ___ such that s > r. Exercise Set 1.2 (p. 13+):
Exercises 4, 8, 12.
4.
a. Is 2 {2}?
In the set-rooster notation, all the elements of a set exist between the braces.
As 2 existed in between the braces of a set {2}, then 2 is an element in the set {2}.
The answer is: Yes
b. How many elements are in set {2,2,2,2}?
The set is in set-rooster notation. So, 2,2,2,2 are the elements in the set {2, 2, 2, 2}.
Therefore all of the elements are equal to 2.
The set contains only one element 2.
c. How many elements are in set {0, {0}}?
The set is in set-rooster notation. So, 0,{0} are the elements in the set {0,{0}}. CISC502 From the discussion of (a), obtained 0 {0}.
As 0 is an element in {0} and {0} is a set for itself, both are no the same. That is, 0 {0}.
Therefore 0 and {0} are two different elements for the corresponding set {0,{0}}/
As a deduction, there are 2 elements in the set {0,{0}}. d. Is {0} {{0}, {1}}?
In the set-rooster notation, all the elements of a set exist between the braces.
As {0} and {1} exists between the braces of a set {{0}, {1}}, then {0} and {1} are
The elements in the set {{0}. {1}}.
The answer is Yes.
e. Is 0 {{0}, {1}}?
In the set-rooster notation, all the elements of a set exist between the braces.
As {0} and {1} exist between the braces of a set {{0}. {1}}. Then {0} and {1} are
The elements in the set {{0}, {1}}/
The answer is No.
8. Let A = {c, d, f, g}, B = {f, j}, and C = {d, g}.
Answer each of the following questions. Give reasons for your answers.
a. Is B A? No
b. Is C A? Yes
. Is C C? Yes
d. Is C a proper subset of A? Yes
12. Let S = {2, 4, 6} and T = {1, 3, 5}. Use the set-roster notation to write each of the following
sets, and indicate the number of elements that are in each set.
. SxT
b. T x S
. SxS
d. T x T
Exercise Set 1.3 (p. 21+):
Exercises 2, 4, 15
2.
Let C = D = {-3, -2, -1, 1,2,3} and define a relation S
From C to D as follows: for all (x, y) C x D,
1 a.
b.
c.
d. 1 (x, y) S means that - - - is an integer.
x y
Is 2 S 2? Is - 1S – 1? Is (3, 3) S? Is (3, -3) S?
Write S as a set of ordered pairs.
Write the domain and co-domain of S.
Draw an arrow diagram for S. 4.
Let G = {-2, 0, 2)} and H = {4, 6, 8} and define a relation
V from G to H as follows: For all (x, y) G x H
x-y
(x, y) V means that ------- is an integer.
4 CISC502 a. Is 2 V 6? Is (-2) V (-6)? Is (0) V? Is (2, 4) V?
b. For all elements x in A and y and z in B, if ____ then ____?
15.
Let X = {2, 4, 5} and Y = {1, 2, 4, 6}. Which of the following
Arrow diagrams determine functions from X to Y? Exercise Set 2.1 (p. 37+):
Exercises 2, 4, 13, 22, 26
In each of 1 – 4 represent the common form of each argument using letters to stand for
component sentences, and fill in the blanks so that the argument in part (b) has the same logical
form as the argument in part (a).
2.
a. If all computer programs contain errors, then this program contains an error.
This program does not contain an error.
Therefore, it is not the case that all computer programs contain errors.
b. If ____, then___.
2. Is not odd.
Therefore, it is not the case that all prime numbers are odd.
4.
a. If n is divisible by 6, then n is divisible by 3.
If n is divisible by 3, then the sum of the digits of n is divisible by 3.
Therefore, if n is divisible by 6 then the sum of the digits of n is divisible by 3.
(Assume that n is particular, fixed integer.)
b. If this function is _____ then this function is differentiable.
If this function is ____ then this function is continuous.
Therefore, if this function is a polynomial, then this function ___.
13. ~(p q) v (p v q)
22. p (q r) and (p q) (p r)
25. Sam is an orange belt and Kate is a red belt.
Exercise Set 2.2 (p. 49+):
Exercises 8, 13, 1
[you can solve 18 using either truth tables or by rewriting statements]
Rewrite the statements in 1 – 4 in if-then form.
1. This loop will repeat exactly N times if it does not contain a stop or a go to.
2. ~p q r CISC502 3. p (q r) (p q) r
4.
Exercise Set 2.3 (p. 61+):
Exercises 12, 28, 29, 30, 31, 32, 38(b-c)
Use truth tables to show that the following forms of argument are invalid.
a. p q
b. p q
q
~p
p
~q
(converse error)
(inverse error)
-----------------------------------------------------------------------------------------------------------------If there are as many rational numbers as there are irrational numbers, then the set of all
irrational numbers is infinite.
The set of all irrational numbers is infinite.
The square of this number is not larger than 4.
----------------------------------------------------------------------------------------------------------------------------If at least one of these two numbers is divisible by 6, then the product o these two numbers is
divisible by 6. Neither of these two numbers is divisible by 6. The product of these two numbers is not divisible by 6.
----------------------------------------------------------------------------------------------------------------------------If this computer program is correct, then it produces the correct output when run with the test
data my teacher gave me.
This computer program produces the correct output when run with the test data my teacher gave
me. This computer program is correct.
----------------------------------------------------------------------------------------------------------------------------Sandra knows Java and Sandra knows C++. Sandra knows C++.
----------------------------------------------------------------------------------------------------------------------------If I get a Christmas bonus, I’ll buy a stereo.
If I sell my motorcycle, I’ll buy a stereo. If I get a Christmas bonus or I sell my motorcycle, then I’ll buy a stereo.
-----------------------------------------------------------------------------------------------------------------------You are visiting the island described in Example 2.3.14 and have the following encounters with
natives.
. Another two natives C and D approach you but only C speaks.
C says: Both of us are knaves.
What are C and D?
c. You then encounter natives E and F.
E says: F is a knave.
F says: E is a knave.
Answers:
1. are all true; true 2. are all true; is false 3. Valid; are all true; is true

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