Readings for Session 1 – (Continued)
Sets and OnetoOne Correspondence
Set:
In mathematics, we call collections
of objects sets.
Example: The collection of
numerals that represents the first twelve counting numbers is the
set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.
OnetoOne Correspondence and Equivalence of Sets:
If
the elements of two sets can be paired so that each element is
paired with exactly one element from the other set, then there
is a onetoone correspondence between the two sets. The two sets are
said to be equivalent.
Notation: The sets A and
B are
equivalent and is denoted as A ~ B.
a
b
c


 C D H 

Note:
Here equal and equivalent mean two
different things. Equal sets are equivalent, but equivalent sets
may not be equal. This
was illustrated in the above example where
A ~ B, but
A
≠
B.
Two sets are equal when they have exactly the same elements, and
sets are equivalent when a onetoone correspondence can be set
up between the two sets.
We have shown
a close relationship between the concept of onetoone
correspondence and the idea of the number of elements in a set,
called the cardinality of a set. (See the counting of the
insects above.) This exploration has led us to the following
definitions relating the sets of natural and whole numbers to
sets. Further, we note that this relationship is closely related
to how small children learn to count.
The set of whole numbers
is the set
W
=
{0, 1, 2, 3, …}.
Notation: If a set A is
equivalent to the set {1, 2, 3, …,
N}, we write
n(A)
= N and say “The
cardinal number of set A
is N.”
Also, n(Ø) = 0. The
cardinal number for an empty set is zero.
Example:
When we counted the insects in the above example, we have
shown a onetoone correspondence between the set {1, 2, 3, …,
12} and the set of insects, i.e., we showed the set of insects
and the set {1, 2, 3, ..., 12} are equivalent. We showed the two
sets are equivalent. This means that after we counted the
insects and said there were twelve insects, we were saying that
the cardinal number for the set of insects is 12.
More examples for the cardinal numbers for sets will be given in the
next session.
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