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. ∗ ∗ ∗ ∗ ∗ ∗ ∗
Solution:
Section B: Proofs
∗ ∗ ∗ ∗ ∗ ∗ ∗ Choose 3 of the following 4 questions. ∗ ∗ ∗ ∗ ∗ ∗ ∗ True or false? If true, prove. If false, derive a counterexample.
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.
Solution: False. For example, x _{= }3 and y _{= }5 then x _{+ }y _{= }8 which is even since 8 _{= }2 _{⋅ }4.
Solution: For n _{= }1 we have 2^{1 }_{= }2^{1}^{+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.
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. ∗ ∗ ∗ ∗ ∗ ∗ ∗
Solution:
(a) The truth tables are:
P R Q ∼ [(P ∧ R) ⇒ (∼ Q ∨ R)] (Q∧ ∼ R) ∧ (P ∧ R)
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P Q R ∼ [(P ∧ R) ⇒ (∼ Q ∨ R)] [(P∧ ∼ P) ∧ R] ∧ Q
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Solution:
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).
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