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Show that + : Rxx R rarr R and xx: R xxR...

Show that `+ : Rxx R rarr R and xx: R xxR rarr R ` are commutative binary operations, but -: `Rxx R rarr R and divid: R^(**)xxR^(**)rarrR^(**)` are not commutative.

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Show that ** :Rxx R rarr R given by (a, b) rarr a + 4b^(2) is a binary operation.

Show that **: RxxR rarr R defined by a^(**) b = a + 2b is not commutative.

Show that ""^** : R xxR rarr R given by a^** b rarr a + 2b is not associative.

Show that zero is the identity for addition on R and 1 is the identity for multiplication on R. But there is no identity element for the operations - : R xx R rarr R and divid R^(*) xxR^(*) rarr R.

Show that the function f : R rarr R given by f(x) =x^(3) is injective.

Show that f: R rarr R , f(x) = x/(x^(2)+1) is not one one and onto function.

Show that the VV: RxxR rarr R given by (a, b) rarr max {a, b} and the ^^: RxxR rarr R given by (a, b) rarr min {a, b} are binary operations.

Find gof and fog, if f:R rarr R and g: R rarrR are given by f(x) = cos x and g(x) = 3x^2 . Show that gof ne fog.

Show that the function f: R rarr R , defined as f(x) = x^2 , is neither one-one nor onto.

KUMAR PRAKASHAN-RELATIONS AND FUNCTIONS -Textbook Illustrations for Practice Work
  1. Let P be the set of all subsets of a given set X. Show that cup: P xxP...

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  2. Show that the VV: RxxR rarr R given by (a, b) rarr max {a, b} and th...

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  3. Show that + : Rxx R rarr R and xx: R xxR rarr R are commutative binar...

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  4. Show that **: RxxR rarr R defined by a^(**) b = a + 2b is not commuta...

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  5. Show that addition and multiplication are associative binary operation...

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  6. Show that ""^** : R xxR rarr R given by a^** b rarr a + 2b is not ass...

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

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  8. Show that -a is the inverse of a for the addition operation '+' on R...

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  9. Show that -a is not the inverse of a in N for the addition operation...

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  10. If R1 and R2 are equivalence relations in a set A, show that R(1) cap...

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  11. Let R be a relation on the set A of ordered pairs of positive integers...

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  12. Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9). Let R1 be a relation in X given ...

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  13. Let f: X rarrY be a function. Define a relation R in X given by R = {...

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  14. Determine which of the following binary operations on the set R are as...

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  15. Determine which of the following binary operations on the set R are as...

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  16. Find the number of all one-one functions from set A = {1, 2, 3} to its...

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  17. Let A = {1, 2, 3} Then show that the number of relations containing (1...

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  18. Show that the number of equivalence relation in the set {1, 2, 3} cont...

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  19. Show that the number of binary operations on {1, 2} having 1 as identi...

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  20. Consider the identity function I(N) : N rarr N defined as I(N) (x) = ...

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