for a,b in R define aRb to mean that ab not =0 .Prove or disprove each of the following 1) the relation R is reflexive 2) the relation R is symmetric 3) the relation R is transitive
Answer (1)
Student Expert
11th Nov, 2020
Hello
1. R is reflexive
Now, ab=0
So, a=0 or b=0 (either a or b can be zero but both cannot be zero)
Thus, aRa => a.a=0 => a is not necessarily 0.
So, R is not reflexive.
2. R is symmetric
We know that, ab=ba (communicative property of multiplication)
=> aRb = ab= 0 = bRa.
So, R is symmetric.
3. R is transitive
Let aRb and bRc be arbitrary such that ab=0 and bc=0.
So, in this case b=0, but not a or c.
so, aRc is not true.
Thus, R is not transitive.
Hence, the relation R is symmetric but not reflecive and transitive. It is not an equivalence relation.
Good Luck
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