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UNIVERSITY OF TORONTO
FACULTY OF APPLIED SCIENCE AND ENGINEERING
ECE190F — Discrete Mathematics
Midterm Test
October 24th, 2005
Examiners: John Carter, Paul McCabe
SOLUTIONS
•
This is a “closed book” test; no aids are permitted.
•
No electronic or mechanical computing devices are permitted.
•
Write your answers in the spaces provided.
•
If necessary, use the backs of the pages for rough work.
•
Show all steps and present all results clearly. State any assumptions that you may make.
•
For full credit, answers to counting problems must be expressed as integers.
•
Please write clearly; if we cannot read an answer, we cannot mark it.
•
This test has 11 pages (including this one). Be sure that you have a complete paper.
Name
Student Number
MARKS
Question
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Total
Value
5
4
8
4
5
6
3
4
6
8
10
10
7
6
86
Mark
Enter the frst letter oF
your Family name here.
1
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View Full Document 1. (5 marks)
Let
p
,
q
, and
r
be the following statements.
p
= “It is winter”
q
= “There is snow on the ground”
r
= “It is cold outside”
Write each of the following statements using only the symbols
p
,
q
, and
r
, the negation
symbol
∼
, and the logical connectives
∧
,
∨
,
→
,
↔
, and
⊕
.
(a) It is winter, but there is no snow on the ground.
p
∧∼
q
(b) There is snow on the ground only if it is winter.
q
→
p
(c) If there is snow on the ground, then it is cold outside.
q
→
r
(d) Either there is no snow on the ground or it is not cold outside or it is winter.
∼
q
∨∼
r
∨
p
(e) If it is not cold outside, then either there is no snow on the ground or it is not winter.
∼
r
→
(
∼
q
p
)
2. (4 marks)
Let
p
,
q
, and
r
be statements. Show that (
p
⊕
q
)
→
(
q
∨
r
) is logically equivalent to
∼
p
∨
q
∨
r
.
pqrp
⊕
qq
∨
r
(
p
⊕
q
)
→
(
q
∨
r
)
∼
p
∨
q
∨
r
TTT F
T
T
T
TTF F
T
T
T
TFT T
T
T
T
TFF T
F
F
F
FTT T
T
T
T
FTF T
T
T
T
FFT F
T
T
T
FFF F
F
T
T
Alternatively, (
p
⊕
q
)
→
(
q
∨
r
)
≡∼
((
p
q
)
∨
(
∼
p
∧
q
))
∨
(
q
∨
r
)
≡
((
∼
p
∨
q
)
∧
(
p
q
))
∨
(
q
∨
r
)
≡
((
∼
p
∨
q
∨
q
)
∧
(
p
q
∨
q
))
∨
r
≡
((
∼
p
∨
q
)
∧
(
p
∨
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This homework help was uploaded on 04/19/2008 for the course ECE 190 taught by Professor Carter during the Fall '06 term at University of Toronto Toronto.
 Fall '06
 Carter

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