# Problem understanding relations syntax.

• MHB
• ktri
In summary: To test R for equivalence, you have to decide whether two relations are equal if and only if they have the same elements and the same relations among those elements.To test R for a partial ordering, you would have to say whether $|a-b|<3$ and $|a-c|<3$ imply that $|b-a|<3$.
ktri
I have this question:2. Let R be a relation on Z with $$\displaystyle R = {(a,b) : |a−b| < 3}.$$
(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: $$\displaystyle 4 * 5 = 5 * 4$$
transitive example: $$\displaystyle 2 < 3$$ and $$\displaystyle 3 < 4$$ so $$\displaystyle 2 < 4$$

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
$$\displaystyle |a - b| < 3$$ and $$\displaystyle 3 < |a - b|$$ ? That dosen't seem right to me. I'm really just not sure what I'm comparing to what.

ktri said:
I have this question:2. Let R be a relation on Z with $$\displaystyle R = {(a,b) : |a−b| < 3}.$$
(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: $$\displaystyle 4 * 5 = 5 * 4$$
transitive example: $$\displaystyle 2 < 3$$ and $$\displaystyle 3 < 4$$ so $$\displaystyle 2 < 4$$

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
$$\displaystyle |a - b| < 3$$ and $$\displaystyle 3 < |a - b|$$ ? 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 $|a-c|<3$.

## 1. What is the importance of understanding problem relations syntax?

Understanding problem relations syntax is crucial for identifying the relationships between different elements in a problem. It helps to break down complex problems into smaller, more manageable parts, and allows for a more structured and systematic approach to problem-solving.

## 2. How does understanding problem relations syntax aid in problem-solving?

By understanding problem relations syntax, scientists are able to identify cause-and-effect relationships between different elements of a problem. This allows for the development of hypotheses and the testing of various solutions to the problem.

## 3. Can understanding problem relations syntax improve the accuracy of scientific research?

Yes, understanding problem relations syntax can improve the accuracy of scientific research by helping scientists to identify potential confounding variables and control for them in their experiments. It also allows for a more thorough analysis of the data collected.

## 4. What are some common challenges in understanding problem relations syntax?

One common challenge is the complexity and interconnectedness of many real-world problems, making it difficult to identify and understand all of the relationships between different elements. Another challenge is the use of technical jargon and complex language in scientific literature that may be difficult for non-experts to understand.

## 5. Are there any strategies for improving problem relations syntax understanding?

Yes, there are several strategies that can help improve problem relations syntax understanding. These include breaking down complex problems into smaller parts, using visual aids such as diagrams or flowcharts, and seeking clarification from experts or colleagues when necessary. Additionally, actively practicing problem-solving and critically analyzing scientific research can also improve understanding over time.

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