Let R be the relation on the set of positive integers such that aRb if...
R cannot be reflexive as 'a' and 'b' have to be distinct in aRb. R is symmetric if a and b have a common divisor, then b and a also have. R is not transitive as aRb and bRc doesn't mean aRc. For example 3 and 15 have common divisor, 15 and 5 have common divisor, but 3 and 5 don't have.
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Let R be the relation on the set of positive integers such that aRb if...
Statement: R is the relation on the set of positive integers such that aRb if and only if a and b are distinct and have a common divisor other than 1.
To determine the properties of relation R, let's analyze each property separately.
Reflexivity:
Reflexivity means that every element in the set is related to itself. In this case, a positive integer a cannot be related to itself because the relation R is defined as aRb if and only if a and b are distinct. Therefore, relation R is not reflexive.
Symmetry:
Symmetry means that if a is related to b, then b is also related to a. Let's consider two positive integers, a and b, such that aRb. This means that a and b are distinct and have a common divisor other than 1. However, if we swap the positions of a and b, we still have distinct positive integers with a common divisor other than 1. Therefore, relation R is symmetric.
Transitivity:
Transitivity means that if a is related to b and b is related to c, then a is related to c. Let's consider three positive integers, a, b, and c, such that aRb and bRc. This means that a and b are distinct with a common divisor other than 1, and b and c are distinct with a common divisor other than 1. However, there is no guarantee that a and c are distinct or that they have a common divisor other than 1. Therefore, relation R is not transitive.
Conclusion:
Based on the analysis above, we can conclude that the correct statement about relation R is:
R is symmetric but not reflexive and not transitive. Therefore, the correct answer is option D.
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