Set Theory

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• Set Theory

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Set theory is the branch of mathematics that
studies sets, which are collections of objects.
Although any type of objects can be collected
into a set, set theory is applied most often to
objects that are relevant to mathematics. The
modern study of set theory was initiated by
Cantor and Dedekind in the 1870s. After the
discovery of paradoxes in informal set theory,
numerous axiom systems were proposed in the early
twentieth century, of which the ZermeloFraenkel
axioms, with the axiom of choice, are the most
well known. Set theory, formalized using
first-order logic, is the most common
foundational system for mathematics. The language
of set theory is used in the definitions of
nearly all mathematical objects, such as
functions, and concepts of set theory are
integrated throughout the mathematics curriculum.
Facts about sets and set membership can be
introduced in High school, along with Venn
diagrams, to study collections of commonplace
physical objects. Operations such as set union
and intersection can be studied in this context.
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Organized Counting

• Believe it or not, counting can be quite
  difficult.
• We will limit our counting to only Finite amounts
  (this course used to be called OAC Finite Math)
• For counts that are not finite (called
  In-finite), the tools of calculus are used.

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Determine the number of people who like McDonalds

• Determine the number of license plates in Ontario

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Determine the number of people who like McDonalds
Determine the number of people who like Pizza Hut
Determine the number of people who like Subway
hmmmmwe need a planTo the parking lot!!!!
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• A set is a clearly defined collection of objects
• Use pictures and algebra

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Venn diagrams are pictures that represent
relationships between sets and subsets of
information.Named after the English logician
John Venn (1834-1923)

• Pictorially

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• To represent the UNIVERSE !!!

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• To represent the UNIVERSE !!..

Use a rectangle
U
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• To represent the
• subsets

Use circles or ellipses
U
A
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• Every subset has a compliment

Use a quote
U
A
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It is common for elements to belong to more than
one set.They can be represented by overlapping
subset circles
U
B
A
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Consider a deck of cards ..
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For example If U deck of cardsA set of red
cardsB set of black acesC set of kingsDraw
the Venn Diagram

• U

B
A
c
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Shading allows you to identify a certain region
of the universe

• To communicate which parts need to be shaded,
  there are essentially only 2 operators needed
• INTERSECTION (and) UNION (or)

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UNION (or) The set consisting of all the
elements in A or B

• U

INTERSECTION (and) The set consisting of all the
elements in both A and B
U
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Start with a generic universe

• Shade the following (on half - sheet)

U
B
A
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• Shade the following A

U
B
A
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• Shade the following A B

U
B
A
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• Shade the following A B

U
B
A
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• Shade the following A B

U
B
A
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• Shade the following A B

Also written as (A B)
U
B
A
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• If U x / -5 lt x lt 6, x E I

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• Worksheet 1
• (diagrams/
• Integer example)