Solution:
To determine which of the given relations on the set of real numbers R is an equivalence relation, we need to verify if each relation satisfies the three properties of an equivalence relation: reflexivity, symmetry, and transitivity.
Let's analyze each option:
Option A:
aR1b⇔|a|=|b|
This means that a and b are related if their absolute values are equal.
Reflexive: For any real number a, we have |a|=|a|, so aR1a is true. Hence, R1 is reflexive.
Symmetric: If aR1b, then |a|=|b|. By the property of equality, |b|=|a|, so bR1a is also true. Hence, R1 is symmetric.
Transitive: If aR1b and bR1c, then |a|=|b| and |b|=|c|. By the transitivity of equality, |a|=|c|, so aR1c is true. Hence, R1 is transitive.
Since R1 is reflexive, symmetric, and transitive, it is an equivalence relation.
Option B:
aR3b⇔a divides b
This means that a and b are related if a divides b.
Reflexive: For any real number a, except for zero, a divides itself, so aR3a is true for a≠0. Hence, R3 is not reflexive for all a∈R.
Symmetric: If aR3b, then a divides b. This does not imply that b divides a. Hence, R3 is not symmetric.
Transitive: If aR3b and bR3c, then a divides b and b divides c. This implies that a divides c. Hence, R3 is transitive.
Since R3 is not reflexive and not symmetric, it is not an equivalence relation.
Option C:
aR2b⇔a≥b
This means that a and b are related if a is greater than or equal to b.
Reflexive: For any real number a,a≥a, so aR2a is true. Hence, R2 is reflexive.
Symmetric: If aR2b, then a≥b. However, this does not imply that b≥a. Hence, R2 is not symmetric.
Transitive: If aR2b and bR2c, then a≥b and b≥c. This implies that a≥c. Hence, R2 is transitive.
Since R2 is not symmetric, it is not an equivalence relation.
Option D:
aR4b⇔a<b
This means that a and b are related if a is less than b.
Reflexive: There is no real number a such that a<a. Hence, R4 is not reflexive.
Symmetric: If aR4b, then a<b. This does not imply that b<a. Hence, R4 is not symmetric.
Transitive: If aR4b and bR4c, then a<b and b<c. This implies that a<c. Hence, R4 is transitive.
Since R4 is not reflexive and not symmetric, it is not an equivalence relation.
Therefore, the only equivalence relation among the given options is Option A :
aR1b⇔|a|=|b|
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