For a, b belong to R define aRb to mean that ab is not equal to 0 . (a) The relation R is Reflexive. (b) The relation R is symmetic. (c) The relation R is transitive.
Answer (1)
11th Nov, 2020
Hello student,
1. R is reflexive
Now, ab != 0
So, a!=0 or b!=0 (neither a nor b can be zero)
Thus, aRa => a.a != 0 => a is not necessarily 0.
So, R is reflexive.
2. R is symmetric
We know that, ab=ba (communicative property of multiplication)
=> aRb = ab != 0 = ba = bRa
So, R is symmetric.
3. R is transitive
Let aRb and bRc be arbitrary such that ab != 0 and bc != 0. (...A)
So, in this case b != 0.
Similarly, neither can a = 0 nor c = 0. (from A)
Therefore, aRc != 0
So, aRc is true.
Thus, R is transitive.
Hence, R is an equivalence relation also.
I hope it helps!
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