Section 3 2 More set operations More operations

Section 3. 2 More set operations

More operations Sometimes an operation forms a new set with a different type of objects than in the original sets. Examples. A = {1, 2, 3}, B = {3, 5} are sets of numbers. l A × B = { (1, 3), (1, 5), (2, 3), (2, 5), (3, 3), (3, 5) } is a set of ordered pairs. l P(B) = { {3, 5}, {3}, {5}, { } } is a set of sets

Definitions × B = { (a, b) : a A and b B}. This is called the Cartesian product of A and B. l. A = { S : S A }. This is called the power set of A. l P(A)

Practice Let A = {2, 3, 4} and B = {3, 5}. Determine: l A×B l B×A l (A B) × (A B) l P(A B)

Practice Let A = {a, b, c, d} and B = {b, c, e}. How many elements are in each of the following sets? l A×B l B×A l (A B) × (A B) l P(A) l P(B) l P(A B)

Practice l List the elements which belong to the following set:

Practice l List five elements that belong to the set:

Practice l List five elements that belong to the set:

Definition l. A partition of a set A is a set S of nonempty subsets of A satisfying the following: ¡ If P 1 and P 2 are two different members of S, then P 1 P 2 = Ø. (That is, distinct parts are disjoint. ) ¡ The union of all of the members of S is the entire set A.

Examples and practice Let A = {1, 2, 3, 4, 5, 6}. 1. {{1, 2}, {4, 3}, {5}, {6}} is a partition of A with 4 parts. 2. {{1, 5, 3, 4, 2, 6}} is a partition of A with just 1 part. 3. {{1, 3}, {4, 2}, {3, 5, 6}} is not a partition of A. Why not? 4. {1, 2, {3, 4}, {5, 6}} is not a partition of A. Why not? 5. {{ }, {1, 4, 2}, {3, 5, 6}} is not a partition of A. Why not?

More practice Which of the following are partitions of Z, the set of all integers? For each that is not, say why not. 1. { {3 n + 1 : n Z}, {3 n + 2 : n Z}, {3 n : n Z} } 2. { {2 n + 1 : n Z}, {3 n + 2 : n Z}, {6 n : n Z} } 3. { {3 n + 1 : n Z}, {6 n + 2 : n Z}, {9 n : n Z} } 4. { {4 n + 1 : n Z}, {4 n + 3 : n Z}, {2 n : n Z} }