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A relation $\mathrm{R}$ is defined on the set $\mathrm{Z}$ of integers as follows:
$m R n \Leftrightarrow m+n$ is odd
Which of the following statements is/are true for $\mathrm{R}$ ?
1.R is reflexive
2.R is symmetric
3.$\mathrm{R}$ is transitive Select the correct answer using the code given below:
Options:
$m R n \Leftrightarrow m+n$ is odd
Which of the following statements is/are true for $\mathrm{R}$ ?
1.R is reflexive
2.R is symmetric
3.$\mathrm{R}$ is transitive Select the correct answer using the code given below:
Solution:
2059 Upvotes
Verified Answer
The correct answer is:
2 only
$\because R$ is a relation defined on the set $Z$ of integers as follows:
$\mathrm{mRn} \Leftrightarrow \mathrm{m}+\mathrm{n}$ is odd
(1) Then, $\mathrm{mRm}=2 \mathrm{~m}$ and $\mathrm{nRn}=2 \mathrm{n}$ are not odd multiples of 2 are not odd. Thus, it is not reflexive.
(2) If $\mathrm{m}$ and $\mathrm{n}$ are numberssuch that $\mathrm{mRn} \Leftrightarrow \mathrm{m}+\mathrm{n}$ is odd. Thus, $\mathrm{n} \mathrm{Rm} \Leftrightarrow \mathrm{n}+\mathrm{m}$ is odd.
This relation is symmetric
(3) $\mathrm{mRn}=\mathrm{m}+\mathrm{n}$, if there is third number $\mathrm{p}$ and $\mathrm{n} \mathrm{Rp}=\mathrm{n}+\mathrm{p}$ is odd. (for $\mathrm{ex}: 2+3=5$ is odd $3+4=7$ is
odd. But, $2+4=6$ is notodd.) Then $\mathrm{mRp}=\mathrm{m}+\mathrm{p}$ may not be odd. So, this relation is not transitive.
$\mathrm{mRn} \Leftrightarrow \mathrm{m}+\mathrm{n}$ is odd
(1) Then, $\mathrm{mRm}=2 \mathrm{~m}$ and $\mathrm{nRn}=2 \mathrm{n}$ are not odd multiples of 2 are not odd. Thus, it is not reflexive.
(2) If $\mathrm{m}$ and $\mathrm{n}$ are numberssuch that $\mathrm{mRn} \Leftrightarrow \mathrm{m}+\mathrm{n}$ is odd. Thus, $\mathrm{n} \mathrm{Rm} \Leftrightarrow \mathrm{n}+\mathrm{m}$ is odd.
This relation is symmetric
(3) $\mathrm{mRn}=\mathrm{m}+\mathrm{n}$, if there is third number $\mathrm{p}$ and $\mathrm{n} \mathrm{Rp}=\mathrm{n}+\mathrm{p}$ is odd. (for $\mathrm{ex}: 2+3=5$ is odd $3+4=7$ is
odd. But, $2+4=6$ is notodd.) Then $\mathrm{mRp}=\mathrm{m}+\mathrm{p}$ may not be odd. So, this relation is not transitive.
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