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प्रश्न
Check the injectivity and surjectivity of the following function.
f : N → N given by f(x) = x3
उत्तर
f : N → N given by f(x) = x3
Let f(x1) = f(x2)
∴ x13 = x23
∴ x13 – x23 = 0
∴ `(x_1 - x_2) underbrace((x_1^2 + x_1 x_2 + x_2^2))_(> 0 "for all" "x"_1, "x"_2 "as it's discriminant" < 0)` = 0
∴ x1 = x2
∴ f is injective.
Numbers from codomain which are not cubes of natural numbers are not images under f.
∴ f is not surjective.
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