Question:
Test whether the relation is reflexive ,symmetric ,or transitive on the set specified R={(m,n):m-n≥7}on z
Answer:
1. Let m ∈ Z. Then m – m = 0 is not greater than or equal to 7. Therefore mRm does not holds for all m in Z
and R is not reflexive.
2. Let m, n ∈ Z and mRn hold. m – n is greater than or equal to 7 and n – m is not greater than or equal
to 7. Thus, mRn ≠> nRm and therefore R is not symmetric.
3. Let m, n, p ∈ Z and mRn, nRp both hold. Then m – n and n – p are both is greater than or equal to 7
Therefore m – p = (m – n) + (n – p) is is greater than or equal to 7
Thus, mRn and nRp => mRp and therefore R is transitive.
So, R is not reflexive, not symmetric but transitive.
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