I) Consider the sets A- (1,2) and B-(a,b.c. a) Find the Cartesian Product, AxB.
ID: 3197313 • Letter: I
Question
I) Consider the sets A- (1,2) and B-(a,b.c. a) Find the Cartesian Product, AxB. Call the resulting set C b) List any 3 different subsets of C, such that the subsets have cardinality 1 c) List any 3 different subsets of C, such that the subsets have cardinality 3 AND such that the none of the subsets are functions. Why are the subsets that you listed not functions? d) List any 3 different subsets of C, such that the subsets have cardinality 3 AND such that each of subset is a valid function. Why are the subsets that you listed satisfy the definition of functions? e) Now using C, list the functions that have domain-(1,2) and codomain-(a,b,cj? How many are there? f How many of the functions you found in e) are one-to-one? g) How many of the functions you found in e) are onto?Explanation / Answer
(a) C = {(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)}
(b) Let the subsets be X1,X2,X3
X1= {(1,a)} ; X2= {(2,a)} ; X3= {(1,c)} ;
(c) Let the subsets be X1,X2,X3
X1 = {(1,a),(1,c),(2,a)} ; X2 = {(1,b),(1,c),(2,a)} ; X3 = {(1,b),(2,a),(2,c)}
The above subsets are not functions as the input has two different outputs. For example in X1, for 1 we have two outputs, i.e. a and c
(d) There can not be any subset of cardinality 3 such that it is a function. The main reason is that the mapping here is from A->B, and there are only 2 elements in A. So if we provide 3 elements in the subset, this will turn to one-many, which is not a function