Problem 1: In calculus and other branches of analysis, the notation (a, 8) represents the set {x Î R | a }. Consider all sets of the form (a, 8) where a Î R.
a. What is the union of all of these sets?
b. What is the intersection of all of these sets?
Problem 2: How many non-proper subsets does the set {1, 2, 3, 4, 5, 6} have?
Problem 3: On the set N of positive integers, the relation "is a divisor of" is often indicated by the symbol |;a|b means that a is a divisor of b. Is this relation reflexive? symmetric? transitive? Provide a justification for each of your answers.
Reflexive:
Symmetric:
Transitive:
Problem 4: Let X = {xÎ R | x >=1 } and define a function g: X à R by g(x) = √(x - 1). What is the image of g?
Problem 5: Let g be as defined in Problem 6, and define h: R àR by h(x) = 2x - 1.
(a) What is the domain of ?
(b) What is the codomain of ?
Problem 6: How many different license plates are possible if each plate has two letters followed by four digits, and no digit can be repeated?
Problem 7: An employer has 18 equally qualified applicants for 11 stock clerk positions. In how many ways can she select 11 new employees from the pool of applicants?
Problem 8: How many 8-bit bytes contain an even number of zeros? Explain how you got (or could get) your answer without listing all 256 8-bit bytes and counting.
Problem 9. Construct the truth table for (p Ù r)à (rÚ s).
Problem 10: Restate each of the following sentences in if...then form:
(a) A sufficient condition for collapse of the pressure vessel is submergence to 2500 feet.
(b) Adequate rain is necessary for a good crop.
Problem 11: For each of the following statements, (a) write it in a totally symbolic form, using the notation style inNegating Quantified Statements -Review. Be sure to explicitly address every quantifier implicit in the statement. (b) using as many negation rules as can possibly be applied, write the symbolic form of the negation of the statement, and (c) write the negation in succinct English. (Do not be concerned about whether the statements are true or false; that would be a distraction.)
(a) Every real number has a cube root.
(b) If the sun is shining, then there will be at least one person swimming in the old mill pond.
(c) Every hydrangea is either pink or blue.
Problem 12: A certain graph has 7 vertices. The degrees of the vertices are 3, 8, 2, 7, 4, 6 and 0. How many edges does this graph have?
Problem 13: Draw a directed graph having the following adjacency matrix:
|
A
|
B
|
C
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D
|
E
|
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A
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1
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0
|
0
|
1
|
0
|
|
B
|
1
|
0
|
0
|
1
|
0
|
|
C
|
0
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1
|
1
|
0
|
0
|
|
D
|
0
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0
|
0
|
1
|
1
|
|
E
|
0
|
1
|
0
|
0
|
0
|
Problem 14: A certain tree has 23 vertices. How many edges does it have?