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Fundamentals of Engineering Analysis
EGR 1302 Unit 4, Lecture A
Approximate Running Time is 22 Minutes
Distance Learning / Online Instructional Presentation
Presented by
Department of Mechanical Engineering
Baylor University
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Slide 1
© 2006 Baylor University
Unit 4 - Discrete Mathematics
 Set Theory
 Applications for Determining Probabilities
 Process Evaluation
 Sets, and Logic
Slide 2
© 2006 Baylor University
Set Definitions & Notation
 A Set is a group of things that share a common rule.
S  { Elements }
S  { All Students at Baylor }
S  { All the Cards in a Deck }
Z  { Set of All Integers }
S  {0,1,2,3,4,5,6,7,8,9}
Decimal Number set
A  {0,1}
0 A
Slide 3
© 2006 Baylor University
Binary Number set
1 A
2 A
Rules for Sets
 A Set can be Finite
 A Set can have an Infinite number of Elements
 “Universal” Set Concept
R  {Set of all Real Numbers }  {,0,}
Z  {Set of all Integers }  {,0,}
N   {Set of all Positive Integers }  {1,}
No Zero
N-  {Set of all Negative Integers }  {,1} No Zero
Slide 4
© 2006 Baylor University
General Set Notation
S  { Elements x | Rules for x}
S  {x | x  N  AND 2  x  8}
S  {2,3,4,5,6,7,8}
a Finite Set
A  {x | x  N  AND  4  x  2}
A  {4,3,2,1}
B  {x | x  R AND  3  x  3}
B  {[ 3,3]}
an Infinite Set
a “Closed Infinite Interval”
Slide 5
© 2006 Baylor University
More Rules for Sets
S  {x | x  R AND x  3}
S  {3  x  3}
an Infinite Set
B  {x | x  Z AND x 2  9}
B  {3,2,1,0,1,2,3}
Slide 6
© 2006 Baylor University
Set Mathematics
 Sets are Equal - if all Elements are the same
A  {1,2,3}
B  {3,2,1}
C  A B
C  {1,2,1,3}
Subset:
A B
AB
A is “Contained” in B - A is a “Proper” Subset
B  { All Students at Baylor }
S  { All Baylor Sophomores }
SB
A B
A  B implies A  B
 the “Null” Set
Slide 7
© 2006 Baylor University
S 0
N  Z
ZR
also called the “Empty” set
UNION of Sets
A B
All of A, OR All of B, OR Both
“OR”
A OR B
B  {x | x  R AND 1  x  3}
A  {x | x  R AND 0  x  2}
A
-1
0
1
2
3
B
A B
A  B  [0,3]
Slide 8
© 2006 Baylor University
4
INTERSECTION of Sets
A B
A “intersect” B
Only Elements in A AND B both at the same time
B  {x | x  R AND 1  x  3}
A  {x | x  R AND 0  x  2}
A  B  [1,2]
A
-1
0
1
2
3
B
A B
Slide 9
© 2006 Baylor University
4
“AND”
INTERSECTION of Sets
B  {3  x  3}
A  {4,3,2,1}
A B  ?
A  B  {3,2,1}
CB  0
Slide 10
© 2006 Baylor University
Sets C and B are “Disjoint”
COMPLEMENT of Sets
Given the Universal Set U
A U
A - all Elements NOT in A
U  { All Students at Baylor }
A  { All Baylor Sophomores }
A  " NOT A"  { all Baylor students not sophomores }
“NOT” is also a Boolean Operator
Slide 11
© 2006 Baylor University
Summary of Set Notation
S  { Elements x | Rules for x}
Union - A B -"OR"
Intersecti on - A B -"AND"
Complement - A -"NOT"
Slide 12
© 2006 Baylor University
This concludes Unit 4, Lecture A
Slide 13
© 2006 Baylor University