Question

A certain relation \(\star\) is defined on \(\mathbb{Z}\) by \(x \star y\) if and only if \(xy \ge 0\). For each of the following questions, be sure to provide a proof supporting your answer. a) Is \(\star\) reflexive? b) Is \(\star\) symmetric? c) Is \(\star\) anti-symmetric? d) Is \(\star\) transitive? e) Is \(\star\) an equivalence relation, a partial order, both, or neither? Note: In both LaTeX and the Word equation editor, the \(\star\) symbol can be typeset with the command \verb|\star|. You may keep track of your progress through the assignment using the dropdown list below. This is optional and only for your own benefit (it is not checked by staff). I have this question.

          A certain relation \(\star\) is defined on \(\mathbb{Z}\) by

\(x \star y\) if and only if \(xy \ge 0\).

For each of the following questions, be sure to provide a proof supporting your answer.

a) Is \(\star\) reflexive?
b) Is \(\star\) symmetric?
c) Is \(\star\) anti-symmetric?
d) Is \(\star\) transitive?
e) Is \(\star\) an equivalence relation, a partial order, both, or neither?

Note: In both LaTeX and the Word equation editor, the \(\star\) symbol can be typeset with the command \verb|\star|.

You may keep track of your progress through the assignment using the dropdown list below. This is optional and only for your own benefit (it is
not checked by staff).
I have this question.

Show more…

A certain relation ⋆ is defined on ℤ by

x ⋆ y if and only if xy ≥ 0.

For each of the following questions, be sure to provide a proof supporting your answer.

a) Is ⋆ reflexive?
b) Is ⋆ symmetric?
c) Is ⋆ anti-symmetric?
d) Is ⋆ transitive?
e) Is ⋆ an equivalence relation, a partial order, both, or neither?

Note: In both LaTeX and the Word equation editor, the ⋆ symbol can be typeset with the command .

You may keep track of your progress through the assignment using the dropdown list below. This is optional and only for your own benefit (it is
not checked by staff).
I have this question.

Added by Gregorio N.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Supreeta N

11/09/2023

Step 1

Step 1: A relation $\star$ on a set $S$ is reflexive if $x \star x$ for all $x \in S$. Show more…

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A certain relation $\star$ is defined on $\mathbb{Z}$ by $x \star y$ if and only if $xy \ge 0$. For each of the following questions, be sure to provide a proof supporting your answer. a) Is $\star$ reflexive? b) Is $\star$ symmetric? c) Is $\star$ anti-symmetric? d) Is $\star$ transitive? e) Is $\star$ an equivalence relation, a partial order, both, or neither? Note: In both LaTeX and the Word equation editor, the $\star$ symbol can be typset with the command \star. You may keep track of your progress through the assignment using the dropdown list below. This is optional and only for your own benefit (it is not checked by staff). I have this question.

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00:02 Hello, we are given a relation on the set of rational numbers and the relation is defined by x is related to y if and only if x equal to y plus 6k for some k in integers.

00:18 Note that this condition is equivalent to the condition that x minus y is a integer multiple of 6 or 6 times k for some k belongs to integers.

00:26 Now the first question about this relation is e star reflexive.

00:30 The answer is yes.

00:32 Now x minus x is equal to 0 and 0 we can write as 0 times 6 and 0 belongs to integers.

00:39 So x star x so x is related to x for all x belongs to r.

00:46 Now second question is e star symmetric.

00:49 The answer is yes.

00:50 Suppose x is related to y.

00:52 This implies that x equal to y plus 6k for some k belongs to integers.

00:58 But this implies that y is equal to x minus 6k.

01:01 But this implies that y is equal to x plus 6 times minus k.

01:06 Now as k belongs to integers minus k will belong to integers.

01:10 So this will imply that y is equal to x plus integer multiple of 6.

01:16 So y is related to x.

01:17 Now x is related to y and y is related to x.

01:20 And this holds good for arbitrary pairs of elements in q.

01:26 So x star y is symmetric.

01:30 Now the next question is is the relation anti -symmetric.

01:36 The answer is no.

01:38 Note that 6 is related to 0 as 6 is equal to 0 plus 1 times 6.

01:44 And 0 is related to 6 as star is arbitrary.

01:49 Sorry star is symmetric.

01:52 But we can directly also we can again prove it also that 0 equal to 6 plus 6 times minus 1...

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