Respuesta :
A bijection between two sets is constructed below and also proved that it is both one-one and onto.
What is meant by one-one function?
The phrase "one-to-one function" should not be mistaken with "one-to-one correspondence," which is used to describe bijective functions, in which each element in the codomain is a precise mirror image of a single element in the domain.
Any function that can work with the operations of two algebraic structures is said to be a homomorphism between them. An injective homomorphism is also referred to as a monomorphism for any commonly occurring algebraic structures, particularly vector spaces.
Let t∈T
Then,
t=2k+1, for some k∈Z
Let s=3k
Then,
s∈S and f(s)=2(s/3)+1
=2k+1
=t
Therefore, for every t∈T there exists s∈S such that f(s)=t
Thus, f is onto.
Hence,
f:S ----> T is a bijective mapping
⇒|S|=|T|
Now, we have to define a mapping f:S ---?T
By f(x)=2(x/3)+1, ∀x∈S
Given,
S={3k|k∈Z} and T={2k+1:k∈Z}
We have to prove:
|S|=|T|
Let x, y∈ S with f(x)=f(y)
Then,
2(x/3)+1=2(y/3)+1
2(x/3)=2(y/3)
(x/3)=(y/3)
x=y
So, f is one-one
Now, we have proved that f is both one-one and onto.
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