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Qu 01: Model the following situations as (possibly weighted, possibly directed) graphs. Draw each graph, and give the corresponding adjacency matrices.
(i) Rokey and Jane are friends. Rokey is also friends with Kevin and Ferdy. Jane ,Kevin and kanny are all friends of each other.
Let R be the relation on Z
(the set of integers) defined by
(x, y) R iff x
2 + y2 = 2k for some integers k 0.
Question 15
R is not antisymmetric. Which of the following ordered pairs can be used together in a
counterexample to prove that R is not antisymmetric? (Remember that R is defined on Z
.)
1. (–1, 1) & (1, –1)
2. (5, 9) & (13, 15)
3. (8, 7) & (7, 8)
4. (3, 1) & (1, 3)
Let R be the relation on Z
(the set of integers) defined by
(x, y) R iff x
2 + y2 = 2k for some integers k 0.
Answer questions 13 to 15 by using the given relation R.
Question 13
Which one of the following is an ordered pair in R?
1. (1, 0)
2. (2, 9)
3. (3, 8)
4. (5, 7)
Question 14
R is symmetric. Which one of the following is a valid proof showing that R is symmetric?
1. Let x, y Z be given.
Suppose (x, y) R
then x
2 + y2 = 2k for some k 0.
ie y
2 + x2 = 2k for some k 0.
thus (x, y) R.
2. Let x, y Z be given.
Suppose (x, y) R
then x
2 + y2 = 2k for some k 0.
ie y
2 + x2 = 2k for some k 0.
thus (y, x) R.
3. Let x, y Z
be given.
Suppose (x, y) R
then x
2 + y2 = 2k for some k 0.
thus (y, x) R.
4. Let x, y Z be given.
Suppose (x, x) R
then x
2 + y2 = 2k for some k 0.
ie y
2 + x2 = 2k for some k 0.
thus (y, y) R.
Mary Ann pays a monthly fee for her cell phone package which includes 700 minutes. She gets Billed an additional charge for every minute she uses the phone past the 700 minutes. During her first month, Mary Ann used 26 additional minutes and her bill was $37.79. During her second month, Mary Ann used 38 additional minutes and her bill was $41.39.
i. Write a function that represents the total monthly cost C(x) of Mary Ann’s cell phone package, where x is the number of additional minutes used.
ii. Find the inverse function
iii. What do x and C-1(x) represent in the context of the inverse function?
iv. How many additional minutes did Mary Ann use if her bill for her third month was $48.89?
Define intersection of two Fuzzy set with example.
Un+2-5Un+1-6Un=4^n+n
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Question 3: Mathematical proficiency and the construction of mathematics ideas.
To answer this question, you need to complete the table below, and also study paragraphs
2.7-2.13 in the study guide: the most important is that you understand what are
• constructivism and behaviourism;
• inductive and deductive thinking;
• instrumental and relational understanding;
• conceptual and procedural knowledge; and
• elements of mathematics proficiency.
Complete the table below:
a) How would a constructivist teacher explain (2)
b) Label the following two statements: inductive reasoning or deductive reasoning:
• 17 and 5 are prime numbers, so all odd numbers are prime numbers
• The interior angles of any triangle add up to 180˚, so a right triangle’s two other
angles are both acute, because they add up to 90˚
inductive reasoning or deductive reasoning:
• 17 and 5 are prime numbers, so all odd numbers are prime numbers
Let X = {2, 3, 6, 12, 24, 36}and relation be such that "x y" if x divides y then draw a Hasse diagram of (X, ).
Let R be a relation on the set of all non-negative integers defined by aRb if and only if a3 - b3 is divisible by 6. Then
