Module Code MA1032N: Logic Lecture for Week Autumn

Agenda Week 6 Lecture coverage: –Power Set –Cartesian Product –Partitions

Power Set The set whose elements consist of all the subsets of a given set A is called the power set of A. This set is written P(A). Thus P(A) = {X:X ⊆ A }

Power Set (Cont.)

Power Set (Cont.) Example: So If B = 2 then P(B) = 4 =2 2

Power Set (Cont.)

Power Set (Cont.)

Power Set (Cont.) Theorem 2 A set containing n distinct elements has 2 n subsets More formally:

The Cartesian Product of Two Sets

The Cartesian Product of Two Sets (Cont.)

The Cartesian Product of Two Sets (Cont.)

Partitions

Partitions (Cont.)

Set Partition In this diagram, the set A (the rectangle) is partitioned into sets W,X, and Y.

Partitions (Cont.)

Partitions (Cont.)

Partitions (Cont.) We implied in our definition of partition that the number of blocks in a partition is finite. A more general definition would allow for an infinite number of blocks, although we will not be concerned with these. However: