Question

If A = {12, 8, 7, 6} and B = {4, 3, 2} is there a surjection from A to B (yes or no)?

If A = {12, 8, 7, 6} and B = {4, 3, 2} is there an injection from A to B (yes or no)?

Define an injection from Z to N in Python or C++.

Answer 1

Let F : X->Y be a function.Then function F is injective if distinct elements of X are mapped to distinct elements of Y.

or we can say, if F(x1)=F(x2), then x1=x2.

Let F : X->Y be a function.Then F is surjective if every element of Y is the image of at least one element of X i.e. image(F)=Y.

Given:

A={12,8,7,6} and B={4,3,2}, So according to definition of surjection we can say that yes, there is a surjection from A to B. But there is not injection from A to B.

Following is an injection from Z to N defined in C++.

Let's assume that elements of Z are z1,z2,z3.......,zn and elements of N are n1,n2,n3,..........,nn. and Z[i] denotes the ith element of Z and N[i] denotes the ith element of N. F is some function that return some function value after calculaton.

So, according to the definition of injection, C++ definition is as follow:

int F(int x){
/*
....  
  
*/
return func_value; //returning some function value.
}
void define_injecton(int Z[],int N){
int n1=Z.size();
int n2=N.size();
int i,j;
bool flag=true;
for(i=1;i<=n1;i++){

for(j=1;j<=n2;j++){

if(F(Z[i])==F(N[j]&&Z[i]!=N[j]){ //condition of injection is false.
flag=false;
return flag; //returning false.
break;
}
}
}
return flag;
}