Let B be a set with B k elementof N If for every function f

Let B be a set with |B| = k elementof N. If, for every function from a set A to B, there must be at least k elements of A having the same image in B, then what is the minimum possible cardinality of A?

Solution

here N is the set of non-negative natural numbers {1,2,3,4,5...} and some times {0,1,2,3,4,5,...}.

>>>Here B = {k natural elements} and A = {has natural elements}.

>>>Image of a set : here we have sets A and B and B must have image of A means all the elements of set A must be in set B and set B may have extra elements also.

>>>Cardinality: for example if set A has a,b elements and set B has a,b elements then cadinality of A and B is represented as AxB = {(a,a),(a,b),(b,a),(b,b)} which is 4.

>>> Therefore in our case B has some k Natural numbers and A must also has k natural numbers and may be less than the B.

|A| <= |B| this is all the possible set for cardinality

>>> but we are asked to find the minimum cardinality A which is |A| x |A| always.