- #1
ironman1478
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Homework Statement
Let A = {1, 2, 3, 4}
Let G = {(4,2), (4,1), (4,3), (2,1), (2,3), (1,3)}
Is G a relation? is G an order relation?
Homework Equations
i think i should put the definitions of a relation and an order relation here. also this is from the book Elementary geometry from an advanced standpoint 3rd edition. section 3.2.
The definition of a relation is
A relation defined on a set A is a subset of AxA (A^2)
or in other words, a relation * is defined by the ordered pairs within a given set, G, which is a subset of AxA
example:
A = {1,2,3}
* = {(1,2), (1,3), (2,3)}
because * is a subset of AxA (which is basically all 9 combinations of 1, 2, 3 in ordered pairs)
* is a relation.
in this case * is actually the < relation
a relation * is a an ordered relation if it satisfies these two conditions
1
for every (a,b) only one of these conditions can be satisfied
a*b, a = b, b * a
2
if a * b and b * c then a * c
The Attempt at a Solution
So i know that it is a relation because G is a subset of AxA, which is all 16 combinations of 1, 2, 3, 4
however i am unsure of how i would determine whether or not its an order relation. i am leaning towards "no" since if G was > then it would only contain {(a,b)|a > b} however, (2,3) contradicts that. it also can't be < since that would only contain {(a,b)|a<b} however because it contains (4,1) that can't be true either. therefore i want to say it isn't an order relation, but i am not sure if this is sufficient proof or not, or if i even approached it correctly.