Answer to Question #316056 in Discrete Mathematics for Alamanda

Question

1. Determine whether these functions are onto, one-to-one, both or neither. Justify your answer.

a)   f(1) = c, f(2) = d, f(3) = a, f(4) = d, f(5) = b

b)   g(1) = d, g(2) = c, g(3) = a, g(4) = b

c)   h(1) = b, h(2) = d, h(3) = b, h(4) = c

Answer 1

$a:\\not\,\,one-to-one, f\left( 2 \right) =f\left( 4 \right) \\onto, f\left( \left\{ 1,2,3,4 \right\} \right) =\left\{ a,b,c,d \right\} \\b:\\one-to-one, g\left( 1 \right) ,g\left( 2 \right) ,g\left( 3 \right) ,g\left( 4 \right) \,\,all\,\,different\\onto, g\left( \left\{ 1,2,3,4 \right\} \right) =\left\{ a,b,c,d \right\} \\c:\\not\,\,one-to-one, h\left( 1 \right) =h\left( 3 \right) \\not\,\,onto, h\left( \left\{ 1,2,3,4 \right\} \right) =\left\{ b,c,d \right\} \ne \left\{ a,b,c,d \right\}$