Answer to Question #278860 in Discrete Mathematics for Azka

Question

For each of the following sets,

a) S={1,2,3}, T ={a, b, c}

b) S={a, b}, T ={1,2,3,4}

c) S={1,2,3,4}, T ={a, b}

Determine whether

1. There is a one-to-one function f:S→T;

2. There is an onto function f:S→T; and

3. If there is a bijective function f:S→T.

4. For each (1–3) if such a function exists, explicitly give it. If no function exists give a short explanation?

Answer 1

1. f : S→T - one-to-one function

a) f(1) = a

f(2) = b

f(3) = c

b) f(a) = 1

f(b) = 2

c) There isn't a one-to-one function, because S has more elements than T.

2.f : S→T - onto function

a) f(1) = a

f(2) = b

f(3) = c

b) There isn't a onto function, because S has less elements than T.

c) f(1) = a

f(2) = a

f(3) = b

f(4) = b

3.f : S→T - bijective function

a) f(1) = a

f(2) = b

f(3) = c

b) and c) There isn't bijective function, because S and T have different numbers of elements.