Transcript Set Theory and Logic - Prairie Spirit Blogs

Foundations 30
Mrs. Wirz
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Introduction – Venn Diagrams
All students in Mrs. Wirz’s class
Students that play sports
Students that have a part-time job
Set: A collection of distinguishable objects for
example, the set of whole numbers is
w={1,2,3,…}
 Element: An object in a set, for example 3 is an
element of D, the set of all digits.
 Universal Set: The set of all elements being
considered, for example the universal set of
digits is D={1,2,3,4,5,6,7,8,9}
 Subset: A set whose elements all belong to
another set, for example, the set of odd digits,
O={1,3,5,7,9} is a subset of D.
In set notation this is written as O D.
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Complement: All the elements that belong to a
universal set, but do not belong to a subset.
For example, the complement of subset O is
O’={2,4,6,8}, which happens to be all the even
digits.
 Empty set: A set with no elements. The set is
denoted as { } or .
 Disjoint: Two or more sets having no elements in
common; for example, the set of even numbers
and the set of odd numbers are disjoint.
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A.
List the elements of the universal set of
students in Foundations 30.
𝐹 = {𝑇𝑒𝑛𝑒𝑖𝑙, 𝐿𝑜𝑔𝑎𝑛, 𝐷𝑎𝑒𝑔𝑎𝑛, 𝐾𝑎𝑦𝑙𝑒𝑖𝑔ℎ, 𝑇𝑎𝑦𝑙𝑜𝑟, 𝑅𝑎𝑐ℎ𝑒𝑙, 𝐽𝑎𝑟𝑒𝑑,
𝑅𝑖𝑙𝑒𝑦, 𝐾𝑎𝑡𝑟𝑖𝑛𝑎, 𝐾𝑎𝑙𝑒, 𝑅𝑢𝑠𝑠𝑒𝑙, 𝐾𝑜𝑙𝑡𝑜𝑛, 𝐿𝑜𝑔𝑎𝑛, 𝐷𝑎𝑤𝑠𝑜𝑛,
𝑇𝑎𝑛𝑛𝑒𝑟, 𝑁𝑖𝑐𝑜𝑙𝑒, 𝑀𝑎𝑡𝑡ℎ𝑒𝑤, 𝐶𝑎𝑠𝑒𝑦, 𝐷𝑖𝑥𝑜𝑛, 𝐸𝑚𝑚𝑎, 𝑅𝑦𝑎𝑛, 𝐴𝑚𝑒𝑒, 𝐶𝑜𝑙𝑒}
F
S
W
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F is the universal set.
W and S are subsets of F.
Write W in set notation.
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W’ is the complement of W
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 Describe what W’ contains
 Write W’ in set notation
What would make an empty set?
Example 1:
A. Using set notation, define the universal set,
S, of all natural numbers 1 to 500.
B. Using set notation define, the subsets, F and
T, which indicate the multiples of 5 and 10,
from 1 to 500.
C. Represents the sets and subsets using a
Venn diagram.
Complete example 2 and 3 on handout.
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Finite Set: A set with a countable number of
elements; for example, the set of even
numbers less than 10, E+{2,4,6,8} is finite.
Infinite Set: A set with infinite number of
elements; for example the set of natural
numbers, N={1, 2, 3,…}
Mutually Exclusive: Two or more events that
cannot occur at the same time; for example,
the Sun rising and the Sun setting are
mutually exclusive events.
Sets are equal if they contain exactly the same
elements, even if the elements are listed in different
orders.
 You may not be able to count all the elements in a
very large or infinite set, such as the set of real
numbers.
 The sum of the number of elements in a set and its
complement is equal to the number of elements in the
universal set
 𝑛 𝐴 + 𝑛 𝐴′ = 𝑛(𝑈)
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When two sets A and B are disjoint,
𝑛 𝐴 𝑜𝑟 𝐵 = 𝑛 𝐴 + 𝑛(𝐵)
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Text: page 154-157
Do # 2,5,8,11
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Sets that are not disjoint share common
elements.
Each area of a Venn diagram represents
something different.
When two non-disjoint sets are represented
in a Venn diagram, you can count the
elements in both sets by counting the
elements in each region only once.
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Each element in a universal set appears only
once in a Venn diagram.
If an element occurs in more than one set, it is
placed in the area of the Venn diagram where
the sets overlap.
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Page 160-161 (on the back of this handout)
# 1, 3
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In set notation, 𝐴 ∩ 𝐵is read as “intersection
of A and B.” It denotes the elements that are
common to both A and B. The intersection is
the region where two sets overlap in the Venn
diagram.
U
A
B
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𝐴 ∪ 𝐵 is read as “union of A and B.” It denotes
all elements that belong to at least one of A
or B.
U
A
B
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A\B is read as “ A minus B.” It denotes the set
of elements that are in set A but not in set B.
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Do Example 1
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The number of elements in the union of two
sets is equal to the sum of the number of
elements in each set, less the number of
elements in both sets.
𝑛 𝐴∪𝐵 =𝑛 𝐴 +𝑛 𝐵 −𝑛 𝐴∩𝐵
Do example 2
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http://www.youtube.com/watch?v=s8HfMhK2g0
Page 172- 174, #1, 6,8,11
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Computer Science
• Software
• Internet searches
• Networking
• Programing
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Electrical Engineering
Geometry and Topology
Biology, Physics, and Chemistry
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Do examples “Investigate the Math,”
example 1,2, and 3.
Do Questions #2,4,6, and 9
Conditional Statement: An “if-then”
statement.
example: “If it is raining outside, then we
practice indoors.”
Hypothesis: “It is raining outside.”
Conclusion: “We practice indoors.”
The hypothesis is the statement that follows
“if,” and the conclusion follows “then.”
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 Case 1: It rains outside, and we practice indoors.
When the hypothesis and conclusion are both true, the
conditional statement is true.
 Case 2: It does not rain outside, and we practice
indoors.
When the hypothesis and the conclusion are both false, then the
conditional statement is true.
 Case 3: It does not rain outside, and we practice outdoors.
When the hypothesis is false and the conclusion is true, the conditional
statement is true.
 Case 4: It rains outdoors, and we practice outdoors.
When the hypothesis is true and the conclusion is false, then the
conditional statement is false.
Let p represent the hypothesis and let q
represent the conclusion.
p
q
p→q
True
True
True
False
False
True
False
True
True
True
False
False
Converse: A conditional statement in which the
hypothesis and the conclusion are switched.
example: If the conditional statement is:
If it is raining outside, then we will practice indoors.
The converse would be: If we practice indoors, then
it is raining outside.
A counter-example could disprove this statement.
Such as, you could be practicing indoors because
the field is being used or under repair.
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Biconditional: A biconditional statement is a
conditional statement whose converse is also true. A
biconditional statement is written as “p if and only if
q.”
Example:
conditional statement: If a glass is half-empty, then it is
half-full.
converse: If a glass is half-full, then it is half-empty.
Both the conditional statement and converse are true,
therefore the biconditional statement is:
A glass is half-empty if and only it is half-full.
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Notation for biconditional: p↔q
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Read over the “in summary.”
Do questions 2, 3, 4, 5, 6, 10, and 13 from
pages 204-205
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Inverse: A statement that is formed by
negating both the hypothesis and the
conclusion of a conditional statement;
for example, for the statement “If a number is
even then it is divisible by 2,” the inverse is “If
a number is not even, then it is not divisible
by 2.”
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Contrapositive: A statement that is formed
by negating both the hypothesis and the
conclusion of the converse of a conditional
statement;
For example, for the statement “If a number
is even, then it is divisible by 2,” the
contrapositive is “If a number is not divisible
by 2, then it is not even.”
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Mrs. Wirz said, “If a polygon is a triangle then it has
three sides.”
A.
Is this statement true? Explain.
B.
Write the converse of this statement. Is it true?
Explain.
C.
Write the inverse of this statement. Is it true?
Explain.
D.
Write the contrapositive of this statement. Is it true?
Explain.
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Consider the statement, “If you live in Saskatoon,
then you live in Saskatchewan.”
A.
Is this statement true? Explain.
B.
Write the converse of this statement. Is it true?
Explain.
C.
Write the inverse of this statement. Is it true?
Explain.
D.
Write the contrapositive of this statement. Is it true?
Explain.
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If a conditional statement is true, then its
contrapositive is true, and vice versa.
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If the inverse of a conditional statement is
true, then the converse of the statement is
also true, and vice versa.
Examples – Read to understand examples 1 and
2. Pay special attention to the notation boxes.
Complete the Your Turns for example 1 and 2.
Assignment - #1-8, page 214-215
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Review Assignment
page 220, #1,2,3,4,6,8.
Due: Monday September 24
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Test – Tuesday September 25