Assignment Hippo Loader





Economics 2P30 Examination

Economics 2P30

Foundations of Economic Analysis

Department of Economics

Midterm Examination #1 - Suggested Solutions

Section A: Definitions

∗ ∗ ∗ ∗ ∗ ∗ ∗ Define 4 of the following 5 terms in two sentences or less. ∗ ∗ ∗ ∗ ∗ ∗ ∗

  1. (3%) Tautology
  2. (3%) Union of X and Y
  3. (3%) Proposition
  4. (3%) Power set of X
  5. (3%) Intersection of X and Y

Solution:

  1. A propositional form that is always true.
  2. X Y = {x x X x Y }.
  3. A statement which is either true or false.
  4. The set of all subsets of X.
  5. X Y = {x x X x Y }.

Section B: Proofs

∗ ∗ ∗ ∗ ∗ ∗ ∗ Choose 3 of the following 4 questions. ∗ ∗ ∗ ∗ ∗ ∗ ∗ True or false? If true, prove. If false, derive a counterexample.

  1. (16%) If A B and B C then B A C.

Solution: False. Let A = {1}, B = {1,2} and C = {1,2,3}. Then we have A B and B C. However, A C = {1} and hence B ⊂/ A C.

  1. (16%) If x and y are both odd, then x + y is odd.

Solution: False. For example, x = 3 and y = 5 then x + y = 8 which is even since 8 = 2 4.

  1. (16%) ∀n ∈ N, 2 + 22 + 23 + ⋯ + 2n = 2n+1 −

Solution: For n = 1 we have 21 = 21+1 2 and hence, the statement is true for n = 1. Now assume 2 + 22 + ⋯ + 2n = 2n+1 − 2. We need to show that 2 + 22 + ⋯ + 2n + 2n+1 = 2n+2 − 2. Naturally: 2 + 22 + ⋯ + 2n + 2n+1 = 2n+1 − 2 + 2n+1 = 2(2n+1) − 2 = 2n+2 − 2. Therefore, the statement is true.

  1. (16%) For every set X, X ∈ P(X) and ∅ ∈ P(X).

Solution: True. We need to show that for all sets X, X X and ∅ ⊂ X. For the former, for all x, x X x X is a tautology. Therefore, X, X ∈ P(X). For the latter, x ∈ ∅ is always false. Hence x ∈ ∅ ⇒ x X is always a true statement. Therefore, ∅ ⊂ X.

Section C: Analytical

∗ ∗ ∗ ∗ ∗ ∗ ∗ Choose 2 of the following 3 questions. ∗ ∗ ∗ ∗ ∗ ∗ ∗

  1. (20%) Suppose P, Q and R are atomic propositions.
    • Derive the truth tables for the following two propositional forms.
  1. ∼ [(P R) ⇒ (∼ Q R)] ii. (Q∧ ∼ R) ∧ (P R)
  • Are the two propositional forms equivalent? Why or why not?
  • Find another propositional form which is equivalent to (i) above.

Solution:

(a) The truth tables are:

P      R     Q        ∼ [(P R) ⇒ (∼ Q R)]         (Q∧ ∼ R) ∧ (P R)

T

T

T

F

F

T

T

F

F

F

T

F

T

F

F

T

F

F

F

F

F

T

T

F

F

F

T

F

F

F

F

F

T

F

F

F

F

F

F

F

  • Yes they are equivalent since their truth tables are identical.
  • For example, the proposition [(P∧ ∼ P) ∧ R] ∧ Q is equivalent since:

P      Q     R       ∼ [(P R) ⇒ (∼ Q R)]          [(P∧ ∼ P) ∧ R] ∧ Q

T

T

T

F

F

T

T

F

F

F

T

F

T

F

F

T

F

F

F

F

F

T

T

F

F

F

T

F

F

F

F

F

T

F

F

F

F

F

F

F

  1. (20%) Let the Universe be U = {1,2,3,4,5,6} and let X = {2,4,5,6} and Y = {1,2,3,4}.
    • Is X Y ? Why or why not?
    • Find X Y .
    • Find Xc and Y c.
    • Find (X Y )c.
    • Is there a relationship between (c) and (d)? Explain in detail.

Solution:

  • No since 5 X but 5 Y .
  • X Y = {2,4}.
  • Xc = {1,3} and Y c = {5,6}.
  • (X Y )c = {1,3,4,5}.
  • By De Morgan’s Law, (X Y )c = Xc Y c.
  1. (20%) Let X, Y , and Z be sets. Assume that all three sets are nonempty. Find a single example for sets X, Y , and Z so that all of the following properties are true. Be clear and make sure you specify the Universe, denoted U. Clearly demonstrate that your example is true.
    • X Y = ∅
    • ((X Y ) ∪ Z) ⊂ U (c) X Z ≠ ∅
    • Y Z ≠ ∅
    • X Y c

Solution: An infinite number of such examples exist. For example, let U = {1,2,3,4},

X = {1}, Y = {3} and Z = {1,2,3}. It follows that X Y = ∅ since there are no elements that are common to both sets ((a) is satisfied). Since X,Y,Z U, it naturally follows that (X Y ) ∪ Z U ((b) is satisfied). X Z = {1} ≠ ∅ ((c) is satisfied). Y Z = {3} ≠ ∅ ((d) is satisfied). Lastly, since X Y = ∅, X Y c immediately follows. One can verify since Y c = {1,2,4,5,6} and since X = {1} it is obvious that X Y c ((e) is satisfied).


Want to order fresh copy of the Sample Economics 2P30 Examination Answers? online or do you need the old solutions for Sample Economics 2P30 Examination, contact our customer support or talk to us to get the answers of it.


Want latest solution of this assignment

AssignmentHippo Features
  • On Time Delivery

    Our motto is deliver assignment on Time. Our Expert writers deliver quality assignments to the students.

  • Plagiarism Free Work

    Get reliable and unique assignments by using our 100% plagiarism-free services.

  • 24 X 7 Live Help

    The experienced team of AssignmentHippo has got your back 24*7. Get connected with our Live Chat support executives to receive instant solutions for your assignment problems.

  • Services For All Subjects

    We can build quality assignments in the subjects you're passionate about. Be it Programming, Engineering, Accounting, Finance and Literature or Law and Marketing we have an expert writer for all.

  • Best Price Guarantee

    Get premium service at a pocket-friendly rate. At AssignmentHippo, we understand the tight budget of students and thus offer our services at highly affordable prices.

Tap to Chat
Get instant assignment help
Tap to Chat
Get instant assignment help