 # Discrete math relation properties. For <= relation on the set of integers, specify if <= is Reflexive(R), Antireflexive(AR), Symmetric(S), Antisymmetric(AS), or Transitive(T). Show your analysis. Quessyrutty6w 2022-07-15 Answered
Discrete math relation properties
I am working on a homework assignment and I am having trouble understanding the problem. I feel as if my professor forgot part of the problem, but I would just like to double check and make sure I am not reading the problem incorrectly. This is the problem:
For $\le$ relation on the set of integers, specify if $\le$ is Reflexive(R), Antireflexive(AR), Symmetric(S), Antisymmetric(AS), or Transitive(T). Show your analysis.
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Step 1
It is reflexive (and therefore not anti-reflexive) since it's true for all elements that $x\le x$ (since $x=x$ is true for all integers). It's not symmetric however, since if $x\le y$ is true then $y\le x$ if false (unless $x=y$).
Step 2
It is anti-symmetric since $\left(x\le y\wedge y\le x\right)\to x=y$ (i.e. x and y are the same integer). Finally, it would be transitive because $x\le y$ and $y\le z$ implies that $x\le z$.

We have step-by-step solutions for your answer! Ibrahim Rosales
Step 1
- The relation $\le$ is reflexive. To see this, consider some $x\in \mathbb{Z}$. Then, $x\le x$, by definition of $\le$.
- By definition of anti-reflexive, $\le$ is not antireflexive.
- The relation $\le$ is not symmetric, as if $x\le y$, $y\nleqq x$.
- The relation is antisymmetric, as if $x\le y$ and $y\le x$, then $y=x$.
Step 2
- This is transitive, as if $x\le y\le z$, then $x\le z$.

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