Assignment #2
I.
True or False
1. Equal sets are always equivalent sets.
2. The set of negative integers are finite.
3. x + 5 > 7 iff x > 2
4. If A = { a, e, i, o, u} then a ∈ A
5. Given the following:
𝐶 = {𝑥 ∈ 𝑍|5𝑥 ≤ 15} 𝑎𝑛𝑑 {𝐷 = 𝑥 ∈ 𝑍|𝑥 2 < 25} 𝑡ℎ𝑒𝑛 𝐶 = 𝐷
6. If 100 is prime, then 7 is not prime.
7. The converse of the statement " If am smart then I am rich" is "If am rich then I am not
smart."
8. Consider the following propositions:
2 ≥ 3 𝑜𝑟 3 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
9. The negation of the statement, "Yesterday was sunny" is "Yesterday was raining."
10. The statement the "P implies Q" does not follow that "Q implies P."
II.
Tell whether the following is well-defined set or not well-defined.
11. Collection of favorite colors.
12. The set of competent Mathematicians in all state colleges and universities in the
Philippines.
13. Collection of brand mobile phones.
III.
Identify the following reasoning as inductive or deductive reasoning.
14. If you brush and floss your teeth daily then you will have fewer cavities. Marie brushes
and flosses her teeth daily. Thus, she will have fewer cavities.
15. It has been observed that earthquakes precede the eruption of volcanoes. Thus,
earthquakes cause volcanoes to erupt.
16. All students go to school. You are a student. Therefore you go to school.
17. Most people are pro-choice on abortion. At least most of the people I know.
IV.
Choose the correct answer from the given choices.
18. Which is the negation of the following statement: "If there is no oxygen, then the candle
will go out".
There is no oxygen, and the candle will not go out.
If the candle will not go out, then there is oxygen.
If the candle will go out, then there is no oxygen.
19. Which is the contrapositive of the statements,
P: The temperature drops below freezing point.
Q: The rain will turn to snow.
If the rain will turn to snow, then the temperature drops below freezing point.
If the temperature does not drop below freezing point, then the rain will not turn to
snow.
If the rain will not turn to snow, then the temperature does not drop below freezing
point.
20. Which is the negation of the statement:
"All mathematicians are ogres."
There is at least one mathematician who is not ogres.
There is at least one mathematician who is an ogres.
There are more than one Mathematician that is an ogre.
21. Choose the pattern that best completes the sequence of matrix.
22. Deanna, Annie, Clayton, Billy, and Elise are all standing in line for the roller coaster.
Deanna is either before Annie or after Elise. The first two spots are filled by boys. Elise is
at the end of the line. If Annie is in front of Elise, then Billy is behind Clayton Who is behind
Billy?
Annie
Clayton
Deanna
Elise
23. Given the statements p, q, r and s.
p: The tour goes to Italy.
q: The tour goes to Spain.
r: We go to Venice.
s: We go to Florence.
Which of the following symbols represent the statement, "If we go to Venice, then we can’t
go to Florence."
p⋀~q
r⋁s
r→~s
24. Given the following sets, what is A' - B ?
U={x│0≤ x ≤ 20}
A={2x│0 ≤ x ≤ 5}
B={2x-1│1 ≤ x ≤ 7}
{6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,20
{0, 2, 4, 6, 8, 10 }
{7}
{12, 14, 15, 16, 17, 18, 19, 20}
25. Given the following sets, what is A U B?
U={x│0 ≤ x ≤ 20}
{0, 1, 2, 3, 4, 5, 6, 7, 8}
A={2x│0 ≤ x ≤ 5}
B={2x -1│1≤ x ≤ 7}
{1, 2, 3, 4, 5, 6, 7, ...20}
{0, 1, 2, 3, 4, 5, 7}
(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13}
26. Choose the pattern that best completes the given sequence of matrix.
27. The following conclusion is based on which assumption:
Gregory had studied all night for his examination.
Gregory received a bad grade on his test.
Gregory didn’t study enough.
Studying more is the only way to improve your score on examination.
Gregory didn’t get enough sleep.
Gregory was paying attention in class.
Gregory studied the right material.
28. Joyce came home late one night to find her neighbor murdered in the kitchen. Which two
statements deductively prove that Joyce is innocent of murder?
i. Joyce loves to have coffee with her neighbor on Sunday morning.
ii. If Joyce works late, she picks up pizza on her way home.
iii. All murderers have been to jail.
iv. If Joyce picks up pizza after work, then she sleeps at her boyfriend’s house.
v. Joyce has never seen a prison cell.
i and ii
iii and iv
iii and v
ii and iv
29. Which of the following numbers can be used to show that if the product of two integers is
positive, their sum is not always positive?
5 and 5
V.
5 and -5
-5 and -5
-1 and 25
Solve and answer the following problems.
30. A family has five children. How many different gender combinations are possible,
assuming that order matters? (For example, having four boys and then a girl is distinct
from having a girl and then four boys). You answer must be numeral only.
31. Express the given set in the Roster method of writing a set D = { all positive integers
between 5 and 10 }.
32. In one of the Colleges of PUP, 100 students were asked about their favorite subjects, the
result is as follows:
55 liked Algebra
48 liked Chemistry
53 liked English
25 liked both Algebra and Chemistry
26 liked both Algebra and English
22 liked both Chemistry and English
10 like all three subjects
Use Venn diagram to illustrate this problem and answer the following questions.
How many students like both Algebra and Chemistry but not English?
33. In problem 26, How many students like both Algebra and English but not Chemistry?
34. In problem 26, How many students like both English and Chemistry but not Algebra?
35. In problem 26, How many students like exactly 1 subject?
36. In problem 26, How many students did not like any subject?
37. Given set A = {all elements that belong to Halogen family}, use roster form to rewrite this
set using chemical symbols only.
38. A young lady has eight shirts and four pairs of pants. How many different outfits can she
make by combining one shirt and one pair of pants?
39. In a class of 25 students, 17 lived with both parents, 21 lived with their mothers, and 20
lived with their fathers. How many lived with neither parent?
40. Emily is thinking of a number. If you multiply her number by 93, add 6, and divide by 3,
you obtain 436. What is her number?
41. Emmanuel is transferring his favorites pet animals, ducks and carabaos to his new farm.
He counts 20 heads and 58 feet. How many ducks and carabaos are in his new farm?
42. Given a Venn diagram, express the shaded portion in terms of symbols, where A and B
are subsets of U.
43. Given a Venn diagram, express the shaded portion in terms of symbols, where A and B
are subsets of U.
44. Translate the following statements into if-then form.
" All prime numbers greater than 2."
45. Translate the following statements into if-then form.
"All good people go to heaven".
46. How many truth possibilities are there with 6 statements.
47. Given the statements p, q, r, s, t and u.
p: The tour goes to Italy.
q: The tour goes to Spain.
r: We go to Venice.
s: We go to Florence.
Represent the following statement in symbols.
"The tour goes to Italy and did not go to Spain."
48. Refer to problem 42 to express the following statement in symbols.
If we go to Venice, then we can’t go to Florence."
49. Refer to problem 42 to express the following statement in symbols.
"We go to Venice or We go to Florence."
50. Refer to problem 42 to express the following statement in symbols.
"If the tour goes to Italy, then we go to Venice."