4.1 Product Sets and Partitions

An ordered pair (a,b) is a listing of the objects a and b in a prescribed order with “a” appearing first and “b” appearing second. The cartesian product AxB is defined as the ordered pairs (a,b) with a ∈ A and b ∈ B

AXB = {(1,r), (1,s),(2,r), (2,s),(3,r),(3,s)} AXB = {(a,b)| } A = {1,2,3} |A| = 3 B = {r,s} |B| = 2 AXB = {(1,r), (1,s),(2,r), (2,s),(3,r),(3,s)} The cardinality of AXB = |A||B| 3*2 = 6 AXB will have 6 elements or pairs a ∈ A and b ∈ B

Partition A partition or quotient set denoted P, of set A is a collection P of non-empty subsets of A such that : 1. Each element of A belongs to a set in P. Each element of P is called a cell or a block 2. If A1 and A2 are distinct elements of P, then A1 ∩ A2 = 0 (A sub 1 intersect A sub 2)

Partition Example A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12

Example A = {a,b,c,d,e,f,g,h} A1 = {a,b,c,d} A2 = {a,c,e,f,g,h} A3 = {a,c,e,g} A4 = {b,d} A5 = {f,h} {A1, A2} IS NOT a partition since A1 ∩ A2 = 0 P = {A3, A4, A5}

Cartesian Plane Example