Download Lesson 86: Greater Than, Trichotomy and Transitive Axioms

Lesson 86:
Greater Than, Trichotomy
and Transitive Axioms,
Irrational Roots
One number is said to be greater
than another number if its graph on
the number line lies to the right of the
graph of the other number. Thus, we
see that -1 is greater than -4,
because the graph of -1 lies to the
right of the graph of -4.
Graphing inequalities on a number
line afford practice with the concept
of greater than and also allows us to
remember the definitions of the
domain and of the subsets of the set
of real numbers. Recall that the
domain of an equation or inequality is
the set of permissible replacement
values of the variable.
Example:
Graph: x < 3; D = {Positive
Integers}
Answer:
See Example
Example:
Graph:
x > -3; D = {Reals}
Answer:
See Example
Some fundamental concepts of
mathematics are difficult to remember
because they are so self-evident. The
trichotomy axiom and the transitive
axiom are good examples. The
trichotomy axiom can be demonstrated
by having someone write a number on a
piece of paper. Then let that person turn
the paper over and write a number on
the other side. There are then exactly
three possibilities.
1. The second number is the same
number as the first number.
2. The second number is greater
than the first number.
3. The second number is less than
the first number.
These statements are self-evident
but are not trivial. They tell us that the
real numbers are arranged in order,
and thus we can say that the real
numbers constitute an ordered set. l
we say that these three statements
form the trichotomy axiom.
Trichotomy comes from the Greek
work trikha, which means “in three
parts”.
The transitive axiom also has three
parts. It is also self-evident but not trivial.
If Arthur is larger than Billy and Billy is
larger than Susan, then Arthur is larger
than Susan. The same statement can be
made using smaller than or the same
size as instead of larger than. Real
numbers are just like people in this
respect, because this thought also
applies to real numbers. We state both
of these axioms formally in the following
box.
Axioms
For any real numbers,
a, b, and c:
Trichotomy
Axiom
Exactly one of the following
is true:
a<b, a = b, or a>b
Transitive
Axiom
If a>b and b>c, then a>c
If a<b and b<c, then a<c
If a=b and b=c, then a=c
Example:
Graph:
x ≤⁄ -4; D = {Negative
Integers}
Answer:
See Example
Example:
Graph:
-x + 4 > 2; D = {Reals}
⁄
Answer:
See Example
Example:
Graph
-x – 4 ≥⁄ -2; D = {Negative
Integers}
Answer:
See Example
Some equations that describe the points
of intersection of a line and a circle
cannot be factored, because the
solutions to these equations are
irrational numbers. These solutions can
be found quickly and easily by using the
quadratic formula. Many people always
use this formula when solving real-life
quadratic equations, because so many
of these equations cannot be solved by
factoring.
Example:
Solve
2
2
x+y=9
y–x=1
Answer:
(-1/2 + √17/2, ½ + √17/2)
(-1/2 − √17/2, ½ − √17/2)
HW: Lesson 86 #1-30