Let R be a relation on Z × Z defined by(a, b)R(c, d) if and only if ad – bc is divisible by 5. Then R isReflexive and symmetric but not transitiveReflexive but neither symmetric not transitiveReflexive, symmetric and transitiveReflexive and transitive but not symmetric
Question
Let R be a relation on Z × Z defined by(a, b)R(c, d) if and only if ad – bc is divisible by 5. Then R isReflexive and symmetric but not transitiveReflexive but neither symmetric not transitiveReflexive, symmetric and transitiveReflexive and transitive but not symmetric
Solution
To determine the properties of the relation R, we need to check the conditions for reflexivity, symmetry, and transitivity.
-
Reflexivity: A relation R is reflexive if for every a in Z, (a, a) R (a, a). For this relation, ad - bc = aa - aa = 0, which is divisible by 5. So, R is reflexive.
-
Symmetry: A relation R is symmetric if for every a, b in Z, if (a, b) R (c, d) then (c, d) R (a, b). For this relation, if ad - bc is divisible by 5, then cd - da = -(ad - bc) is also divisible by 5. So, R is symmetric.
-
Transitivity: A relation R is transitive if for every a, b, c in Z, if (a, b) R (c, d) and (c, d) R (e, f) then (a, b) R (e, f). For this relation, if ad - bc and cf - de are divisible by 5, it does not necessarily mean that af - be is divisible by 5. So, R is not transitive.
Therefore, the relation R is reflexive and symmetric but not transitive.
Similar Questions
n the set N×N, the relation R is defined by (a, b) R(c,d)⇔ad=bc. Then R ispartial order relationequivalence relationreflexive and transitive but not symmetricsymmetric and transitive but not reflexive
Let R be a relation defined on N as a R b is 2a+3b is a multiple of 5,a,b∈N. Then R isnot reflexivetransitive but not symmetricsymmetric but not transitivean equivalence relation
The relation R is defined in the set {1, 2, 3, 4, 5, 6} as R={(a,b):b=a+1}, then R is neither reflexive nor symmetric nor transitiveR is neither reflexive nor symmetric but transitiveR is not reflexive but symmetric and transitiveR is reflexive, symmetric and transitive
A={1,2,3,4}, THEN R={(1,2),(1,3),(3,3),(3,1)} IS A __________ans.TRANSITIVE RELATIONNON SYMMETRIC RELATIONANTI SYMMETRIC RELATIONREFLEXIVE RELATION
Suppose a relation R = {(3, 3), (5, 5), (5, 3), (5, 5), (6, 6)} on S = {3, 5, 6}. Here R is known as _________a.equivalence relationb.symmetric relationc.transitive relation’’d.reflexive relat
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.