Identify the order relation between two integers?

Question

I have to give a 10-15 min teaching demonstration on Monday to my class on this topic and I am not sure what they are asking. Please point me in the right direction.

Answer 1

there are two different order relations between integers.

<:

this relation is antisymmetric
x < x is false for all x
anti-reflexive
is x transitive
id x
<=
this relation is reflexive
x <= x for all x
not symmetric
if x <= y, sometimes y<=x (if they are equal), sometimes not
transitive
if x <= y and y <= z, then x <= z

Answer 2

Your problem is not clearly stated. I think the assignment means "define the relations < and > on the integers, and give examples."

I don't know what level of class you are dealing with, but I suspect it is a mid-level elementary class. If so, I suggest you use the following

DEFINITION. If m and n are integers and m is not equal to n, then we say "m is less that n" (and we write "m < n") if and only if there exists a POSITIVE integer p such that m + p = n. Furthermore, we say "n is greater than m" (and write "n > m") if and only if m < n.

The above definition immediately admits the very impoortant theorem, known as the Trichotomy Law:

THEOREM. If m and n are integers, then one and only one of the following holds:
(1) m = n ,
(2) m < n ,
(3) m > n .

You can easily fill 10 - 15 minutes by giving examples.

CAUTION: There is a logical pitfall here, The above definition assumes the class already knows what a positive integer is, and they can only know that if they already understand something about an "order relation." Namely, n is a positive integer if and only if n > 0.

Good luck.

Answer 3

I'm guessing they're talking about <= or >=. That is the <= operator defines an order relation on the integers, or the reals. An order relation satisfies the properties of reflexivity, antisymmetry, and transitivity.

Reflexivity: For each a, a <= a.
Antisymmetry: For each a and b, if a <= b and b <=a then a = b.
Transitivity: For each a, b and c, if a <= b and b <= c then a <= c.

Answer 4

not sure either
maybe they ask about the following
given two integers find the biggest and the smallest.
You draw the real line, identify the integers, etc