Readings
for Lecture/Lab 1 Sets and Whole Numbers
Ordering the Whole Numbers
What do we mean by 3 < 5? We illustrate with an example.
The three apples can only be set up with a 1-1 correspondence with three of the oranges. Two of the oranges cannot be paired with any apples. We say that the set of apples is equivalent to a subset of the set of oranges.
The above example motivates the following definitions.
Definition.
A set A is a subset of a set B,
denoted A B,
if every element of A is also an
element of B. The set A is a proper subset of set B,
denoted A
B, if A is a subset of B and A is not equal to
set B (written symbolically as A
B if A
B
and A ≠ B).
Examples.
1. Let A
= {1, 3, 5} and B = {1, 2, 3, 4, 5,
6}. Then A B
and A
B
since each element of A is an element of B and 2 is an element of B but
2 is not an element of A.
2. Let C
= {a, b, c} and D = {b, c, d, e}. Then C D. C is not a subset of D since D does not contain the
element a of set C.
3. A
A
and
A. (A set is a subset of itself, and the empty
set is a subset of every set.)
Definition. Let a = n(A) and b = n(B) be whole numbers for finite sets A and B. If A is equivalent to a proper subset of B, then a < b or b > a, these are read as a is less that b and b is greater than a.
Example. Consider our earlier example with the apples and oranges. Since the set of apples is equivalent to a proper subset of the set of oranges, we have that 3 < 5.
