Download File

EXERCISES 1.1
20. The set of people who voted in the last election
Use both the roster and the rule method to specify the sets in
Problems 1 through 10.
1. The RGEP courses you are enrolled in this semester
2. The counting numbers less than 20
3. The fractions whose numerator is 1 and whose
denominator is a counting number less than 12
4. Your siblings
5. The single digits in the Hindu-Arabic system
6. The single Roman numerals
7. The counting numbers between 1 and 20
8. The counting numbers which are multiples of 3 and less
than 20
9. The factors of 124
10. The Philippine presidents
21. Give an example of a set with no element; one element;
two elements; three elements; an infinite number of
elements.
Use the rule method to specify the sets whose elements are
tabulated in Problems 11-14.
11.
12.
13.
For each of the sets listed below, tell which is finite and which is
infinite. For the finite sets, tell which are equivalent and which
are equal.
22. The set of the first 5 counting numbers
23. The set of all counting numbers less than 5
24. The set of all numbers ending in 5
25. The set of distinct letters of the word “fives”
26. The set of points on a given line which are exactly 5 units
away from a given point on the line
For each of the sets listed below (27-30), tell which are
equivalent and which are equal.
27. The set of distinct letters in the word “katakataka”
28. The set
29. The set of distinct letters in the word “tatak”
30. The set
14.
Use the rule method to specify the sets described in Problems 15
to 20 and tell why the roster method is difficult or impossible.
15. The counting numbers greater than 1000
16. The UPLB students who gone abroad
17. The Filipino students who have read Noli Me Tangere
18. The books in the National Library
19. The set of all rectangles whose area is less than 5
31. List all the subsets of
. How many are there? List all
the subsets of
. How many are there?
32. Suppose a set has 5 elements.
a. How many subsets have exactly 1 element? 2
elements? 3 elements? 4 elements? 5 elements?
b. Are there any other subsets?
c. How many subsets does a set of 5 elements have?
j.
33. How many subsets does a set of size n have?
34. Denote by
the set of all subsets of A. If
, find
.
35. If
, what is
?
36. Explain why any subset of a finite set is finite.
37. Can a subset of an infinite set be infinite?
38. Let A be a finite set. Can any of its subsets be equivalent
to it?
39. Let A be an infinite set. Can any of its subsets be
equivalent to it?
40. Algebra of Sets. Draw a Venn diagram to illustrate its
truth or provide a convincing argument.
a.
(Associative Property
of Union)
b.
(Associative Property
of Intersection)
c.
(Distributive
Property of Union Over Intersection)
(Distributive
d.
Property of Intersection Over Union)
e.
(Identity Properties)
f.
(Zero Properties)
g.
(Idempotent Properties)
h.
(Complement Laws)
;
(De
i.
Morgan’s Laws)
(Absorption
Laws)
k.
l.
(Addition Law)
(Simplification Law)
41. Prove the following, using algebra of sets given in
number 40.
a.
b.
c.
d.
e.
42. Define
a. If
.
and
i.
ii.
b. Draw a Venn diagram for
, find
;
.
43. Prove that
and
are disjoint. Prove that
and
are disjoint. Express
as the
disjoint union of sets.
44. If
, find
a.
b.
c.
45. A survey of 100 students gave the following information
about the MST courses they preferred:
40 preferred MATH 1
30 preferred MATH 2
25 preferred NASC 3
20 preferred both MATH 1 and MATH 2
10 referred both MATH 1 and NASC 3
10 preferred both MATH 2 and NASC 3
8 preferred MATH 1, MATH 2, and NASC 3
a.
b.
c.
d.
46. Let
How many preferred MATH 1 but not MATH 2?
How many preferred MATH 2 but not NASC 3?
How many preferred NASC 3 only?
How many did not prefer any of the three
courses?
and
. Find S if
a.
b.
c.
d.
e.
47. If A and B are disjoint, prove that
.
48. Find the following:
a.
e.
b.
f.
c.
g.
d.
h.