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 $|a-c|<3$.
 
Hello, I'm joining this forum to ask two questions which have nagged me for some time. They both are presumed obvious, yet don't make sense to me. Nobody will explain their positions, which is...uh...aka science. I also have a thread for the other question. But this one involves probability, known as the Monty Hall Problem. Please see any number of YouTube videos on this for an explanation, I'll leave it to them to explain it. I question the predicate of all those who answer this...
I'm taking a look at intuitionistic propositional logic (IPL). Basically it exclude Double Negation Elimination (DNE) from the set of axiom schemas replacing it with Ex falso quodlibet: ⊥ → p for any proposition p (including both atomic and composite propositions). In IPL, for instance, the Law of Excluded Middle (LEM) p ∨ ¬p is no longer a theorem. My question: aside from the logic formal perspective, is IPL supposed to model/address some specific "kind of world" ? Thanks.
I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
Back
Top