Answer :
For given function f, f : N² --> Z⁺ be defined as
f(x,y) = 2ˣ3ʸ
Every element of domain(N²) has a unique image in co-domain but for every y∈Z⁺ there does not exit pair (p,q)∈N². Thus, the function f one to one and total but not onto function.
So, Correct option is option(C).
Onto Functions: A function in which every element of Co-Domain Set has one pre-image.
One-to-One Functions: A function in which one element of Domain Set is connected to one element of Co-Domain Set.
Total function: A function which is defined for all inputs of the right type, that is, for all of a domain.
Let f : N² --> Z⁺ be defined as
f(x,y) = 2ˣ3ʸ
then, f is not onto because for z∈ Z⁺
there is (x,y)∈N²
f(x,y) = z is not possible.
f is one to one because for (x₁, y₁) , (x₂,y₂)∈N²
such that f(x₁,y₁) = f(x₂,y₂)
=> 2ˣ¹3zʸ¹= 2ˣ²3ʸ²
=> x₁ = x₂ and y₁ = y₂
=>( x₁ ,y₁) = (x₂, y₂)
so, f is one to one. Every element of domain(N²) has an image in Co-domain(Z⁺). Thus, f is total function.
Hence, f is one to one and total but not onto.
To learn more about One-One and Onto function, refer:
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Final answer:
The function [tex]f(x, y) = 2^(3^x)[/tex] is total, onto, and not one-to-one.
Explanation:
The function [tex]f(x, y) = 2^(3^x)[/tex] is total, onto, and not one-to-one.
Let's break down each statement:
- Total: The function is total because it is defined for every pair of natural numbers (x, y).
- Onto: The function is onto because for every positive integer z, there exists a pair (x, y) such that f(x, y) = z. In other words, every positive integer has a pre-image under f.
- Not one-to-one: The function is not one-to-one because different pairs (x, y) can map to the same positive integer z. Therefore, there are multiple pre-images for some positive integers.
