Function and Relations

Question

I have a small question, if anyone could shed some light I would be really grateful!

we have this relation R: ∀ x,y ∈ "≈" , [(x≈y) ⇔ (| x - y| ≤ 0.5)]

Also relation R belongs to real numbers ℝ.

Is it reflective, symmetric , transitive or anti-symmetric ?

I done some research on Wikipedia and read about these terms and saw the visuals too.

i suppose that x=4.5 and y=5 then:

I think it is both reflective and symmetric.

I also think that it cant be transitive though (say there is a C=5.5, then y≈c but not x≈c) . I have no idea if it is anti-symmetric! This answer came intuitively , but i don't know how can i express this in a formal way!

Any advises are welcome ;)

Answer 1

Hints: you have to find out whether

$$\text{Reflexivity}:\forall\,x,y\in\Bbb R\;,\;\;|x-x|\le\frac12\;\;?$$

$$\text{Symmetry}:\forall\,x,y\in\Bbb R\;,\;\;|x-y|\le\frac12\implies |y-x|\le\frac12\;\;?$$

$$\text{Transitivity}:\forall\,x,y,z\in\Bbb R\;,\;\left(|x-y|\le\frac12\;\wedge\;|y-z|\le\frac12\right)\implies |x-z|\le\frac12\;\;?$$

Funrther hint: work a little harder on the third one above...

Answer 2

Reflexive:

• $\forall x \in \mathbb R, \; |x - x| = 0 \leq 0.5.$
  Therefore, the relation is reflexive.

Symmetric:

• $\forall x, y \in \mathbb R,\;\text{if }\; |x - y| \leq 0.5 \iff |y - x| \leq 0.5$. Hence, $x\approx y \iff y\approx z$.
  Therefore, the relation is symmetric.

Transitive:

• For transitivity, we need for the following to be true:

  $$\forall\,x,y,z \in \mathbb R,\;\text{ if } \; \Big(|x-y|\leq 0.5\;\land\; |y-z|\leq 0.5\Big), \;\text{ then }\; |x-z|\leq 0.5$$ You've provided a nice counterexample: $x = 4.5, y = 5,\; \text{and} \; z = 5.5$. Then we have $x \approx y$ and $y \approx z$, but NOT $x \approx z$.

So the relation is reflexive, symmetric, but not transitive.

Note:

To prove that a relation has a property, you need to show that for arbitrary elements in your set (not specific numbers), the relation holds. That is, we use arbitrary elements (which represent any element in the set) because if a property holds for these arbitrary elements, it holds for all elements.

To prove that a relation does not satisfy a property, it is enough to find and explicitly provide a counterexample: any specific values in the set that shows that the property fails for all elements in the set, thereby proving that the relation does not have that property.