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Let Z be the set of integers. Show that ...

Let `Z` be the set of integers. Show that the relation `R={(a ,\ b): a ,\ b in Z` and `a+b` is even} is an equivalence relation on `Zdot`

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We observe the following properties of `R`.

Reflexivity :

Let a be an arbitrary element of `Z`. Then,

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RD SHARMA-RELATIONS-Solved Examples And Exercises
  1. Prove that the relation R on Z defined by (a ,\ b) in RhArr a-b is di...

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

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

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

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

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

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

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

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

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

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

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  12. Let Z be the set of all integers and Z0 be the set of all non-zero int...

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  13. If R and S are relations on a set A , then prove the following: R and ...

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  14. If R and S are transitive relations on a set A , then prove that RuuS ...

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  15. Write the domain of the relation R defined on the set Z of integers as...

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  16. If R={(x ,\ y): x^2+y^2lt=4; x ,\ y in Z} is a relation on Z , write ...

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  17. Write the identity relation on set A={a ,\ b ,\ c} .

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  18. Write the smallest reflexive relation on set A={1,\ 2,\ 3,\ 4} .

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  19. If R={(x ,\ y): x+2y=8} is a relation on N , then write the range of R...

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  20. If R is a symmetric relation on a set A , then write a relation betwee...

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