List all injective functions from [ 2 ] to [ 3 ].List all surjective functions from [ 3 ] to [ 2 ] .List all bijective functions from [ 2 ] to [ 2 ] . Give a relation that is not a function between [ 2 ] and [ 2 ] and prove that it is not a function.

Question

List all injective functions from   [  2  ] to   [  3  ].List all surjective functions from   [  3  ] to   [  2  ] .List all bijective functions from   [  2  ] to   [  2  ] . Give a relation that is not a function between   [  2  ] and   [  2  ] and prove that it is not a function.

elcotevgs · Accepted Answer

Step 1Let A be a set of cardinal k, and B a set of cardinal n.The number of injective applications between A and B is equal to the partial permutation:             n      !              (      n      −      k      )      !        =            3      !              1      !        =  6The number of surjections between the same sets is   k  !  S  (  n  ,  k  )  k  !  S  (  n  ,  k  ) where   S  (  n  ,  k  )  S  (  n  ,  k  ) denotes the Stirling number of the second kind.A surjection between A and B defines a parition of A in   c  a  r  d  (  B  )  =  k  c  a  r  d  (  B  )  =  k groups, each group being mapped to one output point in B. The number of such partitions is given by the Stirling number of the second kind. Each permutation of these kk defines a different surjection, hence the k! factor.

List all injective functions from [2] to [3]. List all surjective functions from [3] to [2] . List all bijective functions from [2] to [2] . Give a relation that is not a function between [2] and [2] and prove that it is not a function.

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