MHB Problem understanding relations syntax.

  • Thread starter Thread starter ktri
  • Start date Start date
  • Tags Tags
    Relations
AI Thread Summary
The discussion focuses on determining the properties of the relation R defined as R = {(a,b) : |a−b| < 3} on the integers Z. To check if R is reflexive, one must verify that for every integer a, the condition |a-a| < 3 holds true. For symmetry, it needs to be established that if |a-b| < 3, then |b-a| < 3 also holds. Transitivity requires proving that if |a-b| < 3 and |b-c| < 3, then |a-c| < 3 must be true as well. The participant expresses confusion about how to apply these definitions correctly to the relation R.
ktri
Messages
1
Reaction score
0
I have this question:2. Let R be a relation on Z with $$R = {(a,b) : |a−b| < 3}.$$
(1) Is R reflexive? (If yes, prove it; if no, give a counterexample)
(2) Is R symmetric? (If yes, prove it; if no, give a counterexample)
(3) Is R antisymmetric? (If yes, prove it; if no, give a counterexample)
(4) Is R transitive? (If yes, prove it; if no, give a counterexample)
(5) Is R an equivalence relation?
(6) Is R a partial ordering?

My main issue is I'm not sure how to check if R is symetric or transitive etc. I know what those words mean:

symetric example: $$4 * 5 = 5 * 4$$
transitive example: $$2 < 3$$ and $$3 < 4$$ so $$2 < 4$$

but I'm not sure how to determine if R is any of those traits. Like to test if R is reflexive am I checking if
$$|a - b| < 3$$ and $$ 3 < |a - b|$$ ? That dosen't seem right to me. I'm really just not sure what I'm comparing to what.
 
Physics news on Phys.org
ktri said:
I have this question:2. Let R be a relation on Z with $$R = {(a,b) : |a−b| < 3}.$$
(1) Is R reflexive? (If yes, prove it; if no, give a counterexample)
(2) Is R symmetric? (If yes, prove it; if no, give a counterexample)
(3) Is R antisymmetric? (If yes, prove it; if no, give a counterexample)
(4) Is R transitive? (If yes, prove it; if no, give a counterexample)
(5) Is R an equivalence relation?
(6) Is R a partial ordering?

My main issue is I'm not sure how to check if R is symetric or transitive etc. I know what those words mean:

symetric example: $$4 * 5 = 5 * 4$$
transitive example: $$2 < 3$$ and $$3 < 4$$ so $$2 < 4$$

but I'm not sure how to determine if R is any of those traits. Like to test if R is reflexive am I checking if
$$|a - b| < 3$$ and $$ 3 < |a - b|$$ ? That dosen't seem right to me. I'm really just not sure what I'm comparing to what.
Hi ktri, and welcome to MHB!

Reflexive means that each element is related to itself. In the case of this relation R, you have to say whether $|a-a|<3$ (for every integer $a$).

To test R for symmetry, you have to decide whether $|a-b|<3$ implies that $|b-a|<3$.

To test R for transitivity, you have to decide whether $|a-b|<3$ and $|b-c|<3$ implies that $|a-c|<3$.
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
Back
Top