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Enjoy - Finite Math - Quiz, Exercises of Mathematical Methods for Numerical Analysis and Optimization

Baddi University of Emerging Sciences and TechnologiesMathematical Methods for Numerical Analysis and Optimization

Main points of this past exam are: Enjoy, Venn Diagram, Helping People, Sue, Invalid, Argument, Symbolic Statement

Typology: Exercises

2012/2013

Uploaded on 03/31/2013

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MA 110-02
§1.1 – 2.4 Test #1 score
Name:
26 June 2001
1. Use a properly labeled Venn diagram to determine the validity of the following argument.
Explain. (10 points)
1. All politicians enjoy helping people.
2. Sue enjoys helping people.
Therefore Sue is a politician.
Solution:
The argument is invalid. From the Venn dia-
gram, we see Sue can enjoy helping while not
being a politician.
Politicians
those who enjoy
helping people
Sue
2. Construct a truth table to show that the symbolic statement p→qis logically equivalent
to its contrapositive. (10 points)
Solution:
p q p →q˜q→˜p
T T T T
T F F F
F T T T
F F T T
The last two columns agree, so the statement p→qis logically
equivalent to ˜q→˜p.
3. Write the following argument in symbolic form. Then use a truth table to determine if
the argument is valid. (10 points)
If a student studies regularly, then the student does well in school. If the stu-
dent’s teachers are good, then the student does well in school. The student
does not do well in school. Therefore, the student doesn’t study regularly or
the student’s teachers are not good.
Solution:
Use the symbolic representations p: a student studies regularly; q: the student does well in
school; r: the student’s teachers are good. Then the paragraph can be written as
(p →q) ∧(r →q) ∧(˜q)→˜p∨˜r
which turns out to be a tautology, so the argument is valid.
p q r p →q r →q˜q(p →q) ∧(r →q) ∧(˜q) ˜p∨˜rparagraph
T T T T T F F F T
T T F T T T T T T
T F T F F F F F T
T F F F T T F T T
F T T T T F T T T
FT F T T T T T T
FF T T F F F T T
F F F T T T T T T
pf3

Partial preview of the text

Download Enjoy - Finite Math - Quiz and more Exercises Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!

MA 110-

Test

score

Name:

26 June 2001

1. Use a properly labeled Venn diagram to determine the validity of the following argument.

Explain. (10 points)

1. All politicians enjoy helping people.

2. Sue enjoys helping people.

Therefore Sue is a politician.

Solution: The argument is invalid. From the Venn dia- gram, we see Sue can enjoy helping while not being a politician.

Politicians

those who enjoy helping people

Sue

2. Construct a truth table to show that the symbolic statement p → q is logically equivalent

to its contrapositive. (10 points)

Solution: p q p → q (^) ˜ q → (^) ˜ p T T T T T F F F F T T T F F T T

The last two columns agree, so the statement p → q is logically equivalent to ˜ q → (^) ˜ p.

3. Write the following argument in symbolic form. Then use a truth table to determine if

the argument is valid. (10 points)

If a student studies regularly, then the student does well in school. If the stu-

dent’s teachers are good, then the student does well in school. The student

does not do well in school. Therefore, the student doesn’t study regularly or

the student’s teachers are not good.

Solution: Use the symbolic representations p : a student studies regularly; q : the student does well in school; r : the student’s teachers are good. Then the paragraph can be written as ( (p → q) ∧ (r → q) ∧ ( ˜ q)

˜ p^ ∨^ ˜ r

which turns out to be a tautology, so the argument is valid. p q r p → q r → q (^) ˜ q (p → q) ∧ (r → q) ∧ ( ˜ q) (^) ˜ p ∨ (^) ˜ r paragraph T T T T T F F F T T T F T T T T T T T F T F F F F F T T F F F T T F T T F T T T T F T T T F T F T T T T T T F F T T F F F T T F F F T T T T T T

MA 110 Test 1 Solutions page 2

4. Which two of the following statements are logically equivalent? (You don’t need to use a

truth table, but explain why they are in a sentence.) (10 points)

(a) If it is not raining, then I play tennis.

(b) If I play tennis, then it is not raining.

(c) If it is raining, then I don’t play tennis.

(d) I hate tennis, therefore I don’t play tennis.

Solution: Statement (c) is the contrapositive of statement (b), so they are logically equivalent. Statement (b) is the converse of (a) so they aren’t equivalent.

5. If U = { a, b, c, d, e, f , g, h, i, j, k, l, m }, A = { a, c, d, g, j, k, m } and B = { a, c, e, g, i, k, m },

find the set (A ∩ B) ′. Then illustrate (A ∪ B) ′^ by shading the result in a Venn diagram.

(10 points)

Solution: Beginning with the universal set U , just delete all elements that are in A ∩ B to get (A ∩ B) ′^ = { b, d, e, f , g, h, i, l }. The Venn diagram for (A ∪ B) ′^ is shown to the right.

A B

U

6. In a group of 250 students, 165 enjoy attending basketball games, 126 enjoy attending

baseball games, and 61 enjoy neither? How many of the students enjoy both?

Draw a properly labeled Venn diagram and explain your reasoning. (10 points)

Solution: We know that n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Since we are given that 61 like neither, we know that 250 − 61 = 189 like at least one, so n(A ∪ B) = 189. So 189 = 164 + 128 − n(A ∩ B). Thus, n(A ∩ B) = 102.

63 102

61

Ba

ske tball Baseb all

24

7. Compute the numbers 7 P 3 and 7 C 3. Make up two counting problems that would have

these numbers as an answer. (10 points)

Solution: Using the formulas we have 7 P 3 = (^) ( 7 −7! 3 )! = 7 · 6 · 5 = 210 and 7 C 3 = (^) ( 7 −7! 3 ) !3! = 73 ·^6 · 2 ·^5 = 35. For counting problems, you could say:

From a group of 7 people, we want to select 3 and line them up in a row (answer 7 P 3 ). From a group of 7 people, we want to select 3 to form a committee (answer 7 C 3 ).