To determine the properties of the relation $R$ defined on $\mathbb{R}$ by $(x, y) \in R$ if $|x - y| \geq 1$, we will check each property:

(i) Reflexive

A relation is reflexive if for all $x \in \mathbb{R}$, $(x, x) \in R$.

• Here, $|x - x| = 0$, which is not greater than or equal to 1.
• Therefore, $R$ is not reflexive.

(ii) Symmetric

A relation is symmetric if whenever $(x, y) \in R$, then $(y, x) \in R$.

• If $|x - y| \geq 1$, then $|y - x| = |x - y| \geq 1$ as well.
• Therefore, $R$ is symmetric.

(iii) Antisymmetric

A relation is antisymmetric if whenever $(x, y) \in R$ and $(y, x) \in R$, then $x = y$.

• Since $|x - y| \geq 1$ and $|y - x| \geq 1$ both imply $x \neq y$, $R$ cannot be antisymmetric.
• Therefore, $R$ is not antisymmetric.

(iv) Transitive

A relation is transitive if whenever $(x, y) \in R$ and $(y, z) \in R$, then $(x, z) \in R$.

• Consider $x = 0$, $y = 2$, and $z = 3$:

  • $|0 - 2| = 2 \geq 1$ (so $(0, 2) \in R$)
  • $|2 - 3| = 1 \geq 1$ (so $(2, 3) \in R$)
  • But $|0 - 3| = 3 \geq 1$ (so $(0, 3) \in R$).

• However, consider $x = 0$, $y = 2$, $z = 1$:

  • $|0 - 2| = 2 \geq 1$ (so $(0, 2) \in R$)
  • $|2 - 1| = 1 \geq 1$ (so $(2, 1) \in R$)
  • But $|0 - 1| = 1$, so $(0, 1) \notin R$.

• Since we found a counterexample, $R$ is not transitive.

Summary

• Reflexive: No
• Symmetric: Yes
• Antisymmetric: No
• Transitive: No

Would you like more details on any specific property, or do you have any questions? Here are some related questions you might consider:

1. What are examples of reflexive relations on $\mathbb{R}$?
2. Can you provide an example of a relation that is both symmetric and transitive?
3. How do we determine if a relation is an equivalence relation?
4. What other properties can be used to classify relations?
5. How does the definition of $R$ change if the condition is $|x - y| > 1$?

Tip: When analyzing relations, always consider counterexamples to disprove properties like transitivity or antisymmetry.