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Transcript
MTH 231
Section 2.2
Sets, Counting, and the Whole
Numbers
There Are Three Types Of Numbers
1. Nomimal – a sequence of digits used as an
identification (telephone numbers, social
security numbers, account numbers with
stores and banks)
2. Ordinal – represent place in a sequence,
describing the relative position of an object
(first, second, 21st, and so on)
3. Cardinal – answer the question, “How
many?”
An Example
• Blake Sims wore jersey number 6 (nominal) for
the Alabama Crimson Tide last year. He passed
for 3487 yards (cardinal), ranking him second
(ordinal) in the SEC in that category.
More About Sets
• A one-to-one correspondence between sets A
and B is an assignment between the two sets
such that:
1. Every element in A is paired with exactly one
element in B;
2. Every element in B is paired with exactly one
element in A;
3. No element in either set is unpaired.
One-to-One?
1. A classroom with 15 students and 15
textbooks.
2. A classroom with 16 students and 15 books.
3. A classroom with 15 students and 16 books.
The implication of one-to-one is that the two
sets under discussion have the same number of
elements (more on this later).
Why is One-to-One Important?
• Sets A and B are equivalent if there is a one-toone correspondence between A and B.
A~ B
• Moreover, set A is finite if:
1. A is empty, or
2. A is equivalent to a finite set {1, 2, 3, 4,….,n} for
some natural number n.
• A set that is not finite is said to be infinite.
Finite or Infinite?
1. The set of students currently enrolled at
Shelton State
2. The number of grains of sand on the beach
3. Number of votes for Donald Trump in the
2016 Presidential Election
4. The number of numbers between 0 and 1
Cardinality
• Previously, we learned that a cardinal number
can be used to indicate, “How many?”
• The cardinality of a set A—denoted n(A)—is
the number of elements in set A.
• If A is not empty, then n(A) is a counting
number (N, the set of natural or counting
numbers, was defined in Section 2.1). If A is
empty, then let zero (0) be the cardinality of
the empty set.
Finally, The Whole Numbers
• The set of whole numbers, W, is the set of
cardinal numbers of all finite sets:
W = {0, 1, 2, 3, 4, 5,…}
• Another way to think about W:
W  N  0
Whole Numbers and Venn Diagrams
• p. 85, problems 17 through 20.