Answer :
Answer:
a.Neither injective nor surjective
b.Injective but not surjective
c.Surjective but not injective
Step-by-step explanation:
Injective function:It is also called injective function.If f(x)=f(y)
Then, x=y
Surjective function:It is also called Surjective function.
If function is onto function then Range of function=Co-domain of function
a.We are given that
[tex]f:R\rightarrow R[/tex]
[tex]f(x)=x^2[/tex]
It is not injective because
f(1)=1 and f(-1)=1
Two elements have same image.
If function is one-to-one then every element have different image.
Function is not surjective because negative elements have not pre-image in R
Therefore, Co-domain not equal to range.
Given function neither injevtive nor surjective.
b.[tex]f:N\rightarrow N[/tex]
[tex]f(n)=n^2[/tex]
If [tex]f(n_1)=f(n_2)[/tex]
[tex]n^2_1=n^2_2[/tex]
[tex]n_1=n_2[/tex]
Because N={1,2,3,...}
Hence, function is Injective.
2,3,4,.. have no pre- image in N
Therefore, function is not surjective
Because Range not equal to co-domain.
Hence, given function is injective but not Surjective.
c.[tex]f:Z\times Z \rightarrow Z[/tex]
f(n,k)=n+k
It is not injective because
f(1,2)=1+2=3
f(2,1)=2+1=3
Hence, by definition of one-one function it is not injective.
For every element belongs to Z we can find pre- image in [tex]Z\times Z[/tex]
Hence, function is surjective.