Sets.

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Presentation transcript:

Sets

Definition A set is a well-defined collection of objects called its elements. A={1, 2, 3} 1 is in A, 4 is not in A, {3} is not in A

Subsets Set A is a subset of set B if any element in A is also an element in B NEGATION: Set A is not a subset of set B means there is an element in A that is not in B

Venn-Diagrams B A=B A A

Venn-Diagrams B B A A A B

Proper Subset A proper subset of a set is a subset that is not equal to its containing set. Thus B A

Standard Position A B

EXAMPLE 1 Let A = {1} and B = {1, {1}}. a. Is A  B? b. If so, is A a proper subset of B?

EXERCISE (Proofs) Define sets A and B as follows: Is A  B ? c. Is B  A ?

Equality of Sets To know that a set A equals a set B, you must know that A  B and you must also know that B  A.

Example Define sets A and B as follows: Is A = B?

Z, Q, R

Operations Between Sets

Shaded region represents A  B. Shaded region represents A  B. Shaded region represents B – A. Shaded region represents Ac.

EXAMPLE Let the universal set be the set U = {a, b, c, d, e, f, g} and let A = {a, c, e, g} and B = {d, e, f, g}. Find A  B A  B B – A Ac.

There is a convenient notation for subsets of real numbers that are intervals. Observe that the notation for the interval (a, b) is identical to the notation for the ordered pair (a, b). However, context makes it unlikely that the two will be confused.

EXAMPLE Let the universal set be the set R of all real numbers and let These sets are shown on the number lines below. Find A  B, A  B, B – A, and Ac.

THE EMPTY SET

Mutually Disjoint Sets a. Let A1 = {3, 5}, A2 = {1, 4, 6}, and A3 = {2}. Are A1, A2, and A3 mutually disjoint? b. Let B1 = {2, 4, 6}, B2 = {3, 7}, and B3 = {4, 5}. Are B1, B2, and B3 mutually disjoint?

Partition of a Set Suppose A, A1, A2, A3, and A4 are the sets of points represented by the regions shown below. A1, A2, A3, and A4 are disjoint subsets of A, and A = A1 U A2 U A3 U A4.

Let A = {1, 2, 3, 4, 5, 6}, A1 = {1, 2}, A2 = {3, 4}, and A3 = {5, 6}. Is {A1, A2, A3} a partition of A? b. Let Z be the set of all integers and let Is {T0, T1, T2} a partition of Z?

Power Set of a Set Find the power set of the set {x, y}. That is, find

Ordered n-tuples

Cartesian Product

Exercise Let A1 = {x, y}, A2 = {1, 2, 3}, and A3 = {a, b}. a. b. c.