A) reflexive and transitive
B) reflexive and symmetric
C) symmetric and transitive
D) equivalence
Correct Answer: A
Solution :
Given set \[A=\{1,2,3,4\}\]and relation, \[xRy\]if \[x\]divides \[y.\] \[\Rightarrow \]Relation \[=\{(1,1),(2,2),(3,3),(4,4),(1,2),(1,3),(1,4),(2,4)\}\] Reflexive We have, \[xRy\Leftrightarrow y/x\]for \[x,y\in A\] For any \[x\in A,\]we have\[x/x\Rightarrow xRx.\] Thus, \[xRx\]for all \[x\in A.\]So, R is reflexive on A. Symmetry R is not symmetry because, if \[y/x,\]then\[x\]may not divide y. For example 4/2 but 2/4. Transitive, Let \[x,y,z\in A,\]such that \[xRy\]and \[yRz.\] Then, \[xRy\]and \[yRz\Rightarrow \frac{y}{x}\]and \[\frac{z}{y}\Rightarrow \frac{z}{x}.\]So, R is a transitive relation on A.
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