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Prove that the relation congruence modul...

Prove that the relation congruence modulo m on the set `Z` of all integers is an equivalence relation.

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If R is a relation, x R y `\Leftrightarrow` x-y is divisible by m.

x Rx because x-x is divisible by m.

It is reflexive.

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RD SHARMA-RELATIONS-Solved Examples And Exercises
  1. Let N be the set of all natural numbers and let R be a relation on Nxx...

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  2. Let N denote the set of all natural numbers and R be the relation on ...

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  3. Prove that the relation congruence modulo m on the set Z of all int...

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  4. Show that the number of equivalence relations on the set {1, 2, 3} ...

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  5. Given a non-empty set X , consider P\ (X) which is the set of all subs...

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  6. Let R be the equivalence relation in the set A={0,\ 1,\ 2,\ 3,\ 4,\ 5}...

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  7. On the set N of all natural numbers, a relation R is defined as follow...

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  8. Show that the relation R defined by R={(a , b):a-b is divisible ...

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  9. Show that the relation R on the set Z of integers, given by R={(a ,\ b...

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  10. Prove that the relation R on Z defined by (a ,\ b) in RhArr a-b is di...

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  11. Let n be a fixed positive integer. Define a relation R on Z as follows...

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  12. Let Z be the set of integers. Show that the relation R={(a ,\ b): a ,\...

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  13. m is said to be related to n if m and n are integers and m-n is divisi...

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  14. Let R be a relation on the set A of ordered pairs of integers defined ...

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  15. Show that the relation R on the set A={x in Z ;0lt=xlt=12} , given by...

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  16. Let L be the set of all lines in X Y -plane and R be the relation in L...

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  17. Show that the relation R , defined on the set A of all polygons as R={...

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  18. Let O be the origin. We define a relation between two points P and ...

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  19. Let R be the relation defined on the set A={1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7}...

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  20. Let S be a relation on the set R of all real numbers defined by S={(a ...

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