Let f : Z Z be defined as f(x) = x + 3 if x is odd x 5 if x is even . Show that f is a one-to-one correspondence. To show f is a one-to-one correspondence we need to show that f is one-to-one and onto. First show that f is one-to-one. Notice that if f(x) is even, then x must be ---Select--- . Also, if f(x) is odd, then x must be ---Select--- . This is because in both cases of the function definition for f, f(x) differs from x by an ---Select--- integer. An odd integer plus an odd integer is ---Select--- while an odd integer plus an even integer is ---Select--- . Now show that f(a) = f(b) implies a = b for both cases. Suppose f(a) = f(b) is odd. Then a and b are ---Select--- . So a 5 = which simplifies to a = . Similarly if f(a) = f(b) is even then a and b are ---Select--- . Substituting a and b into f we have the equation which simplifies to a = b. Therefore f is one-to-one. To show that f is onto we need to show