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

chillywilly12a 2021-08-13 Answered
Discrete Math Question
Consider the relation R on Z defined by the rule that (a,b)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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Expert Answer

Latisha Oneil
Answered 2021-08-14 Author has 100 answers
Step 1
a) Let a=b=1
Then a+2b=1+2(1)=1+2=3, which is not even, so (1,1)¬{}R
Therefore, for aZ,(a,a)¬{}R
Hence, R is not reflexive.
Step 2
b) Let a=2,b=1
Then a+2b=2+2(1)=4, which is even, so (2,1)R
Now, take a=1,b=2
Then 1+2b=1+2(2)=5, which is not even, so (1,2)¬{}R
Therefore, for a,bZ,(a,b)R but (b,a)¬{}R
Hence, R is not symmetric.
Step 3
c) If (a,b)R and (b,c)R then a and b must be even.
Then the expression a+2c will clearly be even.
Hence, R is transitive.
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