# Discrete Math Question Consider the relation R on Z defined by the rule that (a,b)

Discrete Math Question
Consider the relation R on Z defined by the rule that $\left(a,b\right)\in R$ if and only if $a+2b$ is even. Briefly justify your responses to the following.
a) Is this relation reflexive?
b) Is this relation symmetric?
c) Is this relation transitive?
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Step 1
a) Let $a=b=1$
Then $a+2b=1+2\left(1\right)=1+2=3$, which is not even, so $\left(1,1\right)\mathrm{¬}\left\{\in \right\}R$
Therefore, for $a\in Z,\left(a,a\right)\mathrm{¬}\left\{\in \right\}R$
Hence, R is not reflexive.
Step 2
b) Let $a=2,b=1$
Then $a+2b=2+2\left(1\right)=4$, which is even, so $\left(2,1\right)\in R$
Now, take $a=1,b=2$
Then $1+2b=1+2\left(2\right)=5$, which is not even, so $\left(1,2\right)\mathrm{¬}\left\{\in \right\}R$
Therefore, for $a,b\in Z,\left(a,b\right)\in R$ but $\left(b,a\right)\mathrm{¬}\left\{\in \right\}R$
Hence, R is not symmetric.
Step 3
c) If $\left(a,b\right)\in R$ and $\left(b,c\right)\in R$ then a and b must be even.
Then the expression $a+2c$ will clearly be even.
Hence, R is transitive.