misurrosne

2022-06-25

Working out if a given relation is reflexive, symmetric or transitive (or all 3?)
On the set of integers, let š„ be related to š¦ precisely when
On the set of integers, let š„ be related to š¦ precisely when
1. Is this Reflexive?
2. Is this Symmetric?
3. Is this Transitive?
I'm also wondering if it can be multiple? I assume it can maybe be two but maybe not all 3.
To my understanding:
Reflexive is when each element is related to itself, I am not sure how to apply that to ? (Edit: If $x=3$ and $y=3$, then , so it can't be reflexive as it would be an incorrect statement, so for not equals to it can never be reflexive from what I studied going back over notes)
Symmetric is when x is related to y, it implies that y is related to x (which may be fitting here as x is related to y when they don't equal each other?)
Transitive: When x is related to y, and y is related to z, then x is related to z (Not applicable here? Unsure)
I'm not sure if it's reflex as $xāZ$ and $yāZ$ (both are related to the set of integers), it could be symmetric as they are related when is the same as being related when , then I'm not sure of transitive.

Harold Cantrell

Step 1
You are correct, the relation is not relfexive.
Now, it is time to formally prove that.
To prove that it is not, you must prove that the statement " is a reflexive relation" is false.
First we use the definition of reflexivity to rewrite the above statement into:
Now, we must prove the above statement is false. Since the statement is of the type "$\mathrm{ā}xāX:P\left(x\right)$", it is enough to find one value of x such that P(x) is not true (this value is then called the *counterexample). In your case, taking $x=3$ is perfectly OK, because is false.
Step 2
Alternatively, you could just prove the negation of the statement. The negation is $\mathrm{ā}xāZ:x=x$ this statement can be proven, because $x=3$ satisfies the relation $x=x$.
Step 2
For symmetry, you are correct that the relation is symmetric.
You can do this by proving the statement:
Formally, can prove any statement of the type $\mathrm{ā}x,yāA:P\left(x,y\right)\phantom{\rule{thickmathspace}{0ex}}ā¹\phantom{\rule{thickmathspace}{0ex}}Q\left(x,y\right)$ by:
Taking any two values $x,yāA$
Assuming P(x, y) is true
From that, proving Q(x, y) must also be true.

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