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University of British Columbia
MIDTERM TEST 1: MATHEMATICS
Date: October 28, 2010
Time: 8:30 a.m. to 9:30 a.m.
Number of pages: 10 (including cover page)
Exam type: Closed book
Aids: No calculators or other electronic aids
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Question
Mark
Possible marks
1
7
2
6
3
7
4
5
5
5
6
1 (bonus)
Total
30
1. Consider the following propositions:
W
M
B
Wdd
= “The woman is blue”.
= “The man is blue”.
= “The boy is blue”.
= “The woman has alleles dd ”.
WDd
Mdd
MDd
Bdd
= “The woman has alleles Dd ”.
= “The man has alleles dd ”.
= “The man has alleles Dd ”.
= “The boy has alleles dd ”.
1. Consider the following axioms:
¬W, ¬M, B, Bdd ⇒ (Wdd ∨ WDd ) ∨ (Mdd ∨ MDd ).
1. (a) [2 marks] Provide the smallest possible truth table that shows that these axioms are consistent.
1. (b) [3 marks] Fill in the “Justification” and “Axioms” columns of the proof on the following page. In the
Justification column, provide the Rule of Inference used and the rows to which the rule was applied.
In the Axioms column, provide the list of axioms on which the theorem depends.
1. (c) [1 mark] What does the hypothetical axiom (5) mean, in biological terms?
1. (d) [1 mark] What is the conclusion, in biological terms?
2
Proposition
Justification
Axioms
1.
¬W
Ax
1
2.
¬M
Ax
2
3.
B
Ax
3
4.
(Bdd ∨ BDd ) ⇒ ((Wdd ∨ WDd ) ∨ (Mdd ∨ MDd ))
Ax
4
5.
(B ⇔ (Bdd ∨ BDd )) ∧ ((W ⇔ (Wdd ∨ WDd )) ∧ (M ⇔ (Mdd ∨ MDd )))
HAx
5
6.
B ⇔ (Bdd ∨ BDd )
P5
5
7.
(W ⇔ (Wdd ∨ WDd )) ∧ (M ⇔ (Mdd ∨ MDd ))
P5
5
8.
W ⇔ (Wdd ∨ WDd )
P7
5
9.
M ⇔ (Mdd ∨ MDd )
P7
5
10.
(B ⇒ (Bdd ∨ BDd )) ∧ ((Bdd ∨ BDd ) ⇒ B)
ME 6
5
11.
B ⇒ (Bdd ∨ BDd )
P 10
5
12.
(Bdd ∨ BDd )
13.
((Wdd ∨ WDd ) ∨ (Mdd ∨ MDd ))
14.
(W ⇒ (Wdd ∨ WDd )) ∧ ((Wdd ∨ WDd ) ⇒ W )
15.
(Wdd ∨ WDd ) ⇒ W
16.
¬(Wdd ∨ WDd )
17.
(Mdd ∨ MDd )
18.
(M ⇒ (Mdd ∨ MDd )) ∧ ((Mdd ∨ MDd ) ⇒ M )
P9
5
19.
(Mdd ∨ MDd ) ⇒ M )
P 18
5
20.
M
21.
M ∧ ¬M
22.
¬((B ⇔ (Bdd ∨ BDd )) ∧ ((W ⇔ (Wdd ∨ WDd )) ∧ (M ⇔ (Mdd ∨ MDd ))))
MP
4,
8
P5
5
5
MT
13, 16
MP 17, 19
3
2. (a) [2 marks] Explain, in terms of δ and , what the equation
lim f (x) = L
x→a
means.
2. (b) [4 marks] Prove, using the definition you provided above, that
1
x + 3 = 5.
lim
x→4 2
4
3. (a) [3 marks] Prove that
lim (x − 1) sin
x→1
1
x−1
=0
using appropriate theorems. (You need not prove this in terms of δ and .)
3. (b) [4 marks] Let

1


for x < 1

 (x − 1) sin x − 1
f (x) =
g(x)
for 1 ≤ x ≤ 2 .



 x
e
for x > 2
Find a function g(x) such that f (x) is continuous everywhere. You must justify your answer.
5
4. [5 marks] Two sprinters end a race in a tie. Prove that, at some point in the race, they were running at
exactly the same speed. (Hint: consider the function f (t) − g(t) and its derivative, where f (t) and g(t)
denote the positions of the sprinters relative to the starting line at time t.)
6
5. [5 marks] Where do the maxima of cos
1
x
occur? Justify your answer using appropriate tests.
7
6. [1 bonus mark] Recall that, under the Continuum Hypothesis, the cardinality of the uncountable set of
real numbers R is denoted ℵ1 . Describe a set with “larger” cardinality. You do not need to justify your
answer.
8
This page may be used for rough work. It will not be marked.
9
Table 1: Rules of Inference. The notation A ` B means: from statement A we can infer statement B as
a valid argument. The validity of all these can (and should) be verified using truth tables. Lifted partially
from Wikipedia’s Propositional calculus page.
Rule name
Formalism
Modus Ponens (MP)
((p ⇒ q) ∧ p) ` q
Modus Tollens (MT)
((p ⇒ q) ∧ ¬q) ` ¬p
Disjunctive Syllogism (DS)
((p ∨ q) ∧ ¬p) ` q
Projection (P)
(p ∧ q) ` p
Conjunction (C)
p, q ` (p ∧ q)
Addition (A)
p ` (p ∨ q)
De Morgan’s Theorem (1) (DMT)
¬(p ∧ q) ` (¬p ∨ ¬q)
De Morgan’s Theorem (2) (DMT)
¬(p ∨ q) ` (¬p ∧ ¬q)
Distribution (1) (D)
(p ∧ (q ∨ r)) ` ((p ∧ q) ∨ (p ∧ r))
Distribution (2) (D)
(p ∨ (q ∧ r)) ` ((p ∨ q) ∧ (p ∨ r))
Double Negation (DN)
p ` ¬¬p
Double Negation (DN)
¬¬p ` p
Transposition (T)
(p ⇒ q) ` (¬q ⇒ ¬p)
Material Implication (1) (MI)
(p ⇒ q) ` (¬p ∨ q)
Material Implication (2) (MI)
(¬p ∨ q) ` (p ⇒ q)
Material Equivalence (ME)
(p ⇔ q) ` ((p ⇒ q) ∧ (q ⇒ p))
Rule of Deduction (RD)
(A1 , A2 , A3 , · · · AN , p ` q) ` (A1 , A2 , A3 , · · · AN ` p ⇒ q)
Rule of Contradiction (RC)
(A1 , A2 , A3 , · · · AN , p ` q ∧ ¬q) ` (A1 , A2 , A3 , · · · AN ` ¬p)
(For this last rule, A1 , A2 , A3 , · · · AN must be consistent.)
The following rules can be “skipped steps”:
Commutation (1)
(p ∨ q) ` (q ∨ p)
Commutation (2)
(p ∧ q) ` (q ∧ p)
Commutation (3)
(p ⇔ q) ` (q ⇔ p)
Association (1)
(p ∨ (q ∨ r)) ` ((p ∨ q) ∨ r)
Association (2)
(p ∧ (q ∧ r)) ` ((p ∧ q) ∧ r)
10