Working out if a given relation is reflexive, symmetric or transitive (or all 3?) On the set of int

misurrosne 2022-06-25 Answered
Working out if a given relation is reflexive, symmetric or transitive (or all 3?)
On the set of integers, let 𝑥 be related to 𝑦 precisely when x y
On the set of integers, let 𝑥 be related to 𝑦 precisely when x y
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 x y? (Edit: If x = 3 and y = 3, then x y, 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 x Z and y Z (both are related to the set of integers), it could be symmetric as they are related when x y is the same as being related when y x, then I'm not sure of transitive.
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Answers (1)

Harold Cantrell
Answered 2022-06-26 Author has 21 answers
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: x Z : x x
Now, we must prove the above statement is false. Since the statement is of the type " x X : P ( x )", 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 3 3 is false.
Step 2
Alternatively, you could just prove the negation of the statement. The negation is x 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: x , y Z : x y y y
Formally, can prove any statement of the type x , y A : P ( x , y ) Q ( x , y ) by:
Taking any two values x , y A
Assuming P(x, y) is true
From that, proving Q(x, y) must also be true.

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