Let (x1,y1)~(x2,y2) if (y1-y2)=x1^2-x2^2.
is this reflexive, Symmetrical, Transitive and an Equiv. Relation? if so y?
The above problem is a homework problem that I need to figure out. Any help would be appreciated. Thanks, Ed.
please refrain from using chat talk, that is, type out "why" as opposed to typing "y". this is for two reasons: (1) this is a math forum, we usually use single letters to represent variables or things like that, so it may cause confusion, (2) this is a prestigious math site i'm pushing the envelope enough just by not capitalizing words that should be capitalized and neglecting the proper use of punctuations.
we cannot give you a full solution to homework problems. so you have to be satisfied with hints.
do you know what each of the terms you typed mean? what does it mean to be reflexive? symmetric? transitive? what does it mean to be an equivalence relation? answer those and we'll take it from there
if that's how you define them, then you're going to have a hard time proving these.
A relation $\displaystyle \sim$ on a set $\displaystyle A$ is:
reflexive if $\displaystyle a \sim a$ for all $\displaystyle a \in A$
symmetric if $\displaystyle a \sim b \implies b \sim a$ for all $\displaystyle a,b \in A$
transitive if $\displaystyle a \sim b$ and $\displaystyle b \sim c$ implies $\displaystyle a \sim c$ for all $\displaystyle a,b,c \in A$
and you are correct, the relation is an equivalence relation if it is reflexive, symmetric and transitive
now on your problem. start with reflexivity.
let $\displaystyle A$ be the set the relation is on. we pick some $\displaystyle (x_1,y_1) \in A$. does this point relate to itself? that is, can you show whether or not it satisfies the condition mentioned?
It should be relative because it's a point on a graph, right?
edit:
At least it should be if the points are equal. (1,1) or (2,2). I wouldnt be if the points where (1,3) or (3,6). Is what i'm getting after reading through my book.
no
you must follow the condition they gave you. you can't be guessing randomly like that.
let me do reflexive, you do the others.
we are told that $\displaystyle (x_1,y_1) \sim (x_2,y_2)$ if $\displaystyle \color{red}y_1 - y_2 = x_1^2 - x_2^2$
does this work for a single point? in other words, we want to know if $\displaystyle (x_1,y_1) \sim (x_1,y_1)$.
Let $\displaystyle (x_1,y_1) \in A$
Since $\displaystyle y_1 - y_1 = 0 = x_1^2 - x_1^2$, we have $\displaystyle (x_1,y_1) \sim (x_1,y_1)$. thus, $\displaystyle \sim$ is reflexive