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Show that the relation R on the set Z of...

Show that the relation `R` on the set `Z` of integers, given by `R={(a ,\ b):2` divides `a-b}` , is an equivalence relation.

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Given as,
`R = {(a, b): 2` divides `a – b}` is a relation defined on `Z.`
To prove equivalence relation, the given relation should be reflexive, symmetric and transitive.
We have to check these properties on `R.`
...
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NCERT-RELATIONS AND FUNCTIONS-SOLVED EXAMPLES
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  3. Show that the relation R on the set Z of integers, given by R={(a ,\ b...

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  6. Let A be the set of all students of a boys school. Show that the rela...

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  7. Show that – a is the inverse of a for the addition operation '+' on R ...

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  8. Show that zero is the identity for addition on R and 1 is the identit...

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  9. Show that the vv: R ->R given by (a , b)->m a x {a , b}and the ^^: R -...

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  10. Let P be the set of all subsets of a given set X. Show that uu: P xx ...

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  11. Show that ∗ : RxxR->R given by (a ,b)->a+4b^2is a binary operation.

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  12. Show that subtraction and division are not binary operations on N.

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  13. Show that *: Rxx R ->Rgiven by a*b = a +2bis not associative.

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  14. Show that addition and multiplication are associative binary operatio...

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  15. Show that *: R xxR ->Rdefined by a*b = a +2bis not commutative.

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  16. Show that + : R xx R ->R and xx : R xx R ->R are commutative binary ...

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  17. Let Y = {n^2: n in N} in N. Consider f : N ->Yas f(n)=n^2. Show tha...

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  18. Let f : N ->R be a function defined as f(x)=4x^2+12 x+15. Show that f...

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  19. Consider f : N ->N, g : N ->Nand h : N ->Rdefined asf (x) = 2x, g (y) ...

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  20. Consider f : {1, 2, 3} ->{a , b , c}and g : {a , b , c} ->{a p p l e ,...

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