MATH 105: Finite Mathematics 6-1: Sets Prof. Jonathan Duncan Walla - - PowerPoint PPT Presentation

Introduction to Sets Combining Sets Visualizing Sets Conclusion

MATH 105: Finite Mathematics 6-1: Sets

• Prof. Jonathan Duncan

Walla Walla College

Winter Quarter, 2006

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Outline

1

Introduction to Sets

2

Combining Sets

3

Visualizing Sets

4

Conclusion

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Outline

1

Introduction to Sets

2

Combining Sets

3

Visualizing Sets

4

Conclusion

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Sets and Probability

In order to study probability, how likely an event is to happen, we have to be able to identify collections of possible outcomes. We call these collections sets. Set A Set is an unordered collection of objects. The objects in a set are called elements of the set. Examples License Plates Poker Hands Students in a Class Even Digits

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Sets and Probability

In order to study probability, how likely an event is to happen, we have to be able to identify collections of possible outcomes. We call these collections sets. Set A Set is an unordered collection of objects. The objects in a set are called elements of the set. Examples License Plates Poker Hands Students in a Class Even Digits

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Sets and Probability

In order to study probability, how likely an event is to happen, we have to be able to identify collections of possible outcomes. We call these collections sets. Set A Set is an unordered collection of objects. The objects in a set are called elements of the set. Examples License Plates Poker Hands Students in a Class Even Digits

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Sets and Probability

In order to study probability, how likely an event is to happen, we have to be able to identify collections of possible outcomes. We call these collections sets. Set A Set is an unordered collection of objects. The objects in a set are called elements of the set. Examples License Plates Poker Hands Students in a Class Even Digits

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Sets and Probability

In order to study probability, how likely an event is to happen, we have to be able to identify collections of possible outcomes. We call these collections sets. Set A Set is an unordered collection of objects. The objects in a set are called elements of the set. Examples License Plates Poker Hands Students in a Class Even Digits

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Sets and Probability

In order to study probability, how likely an event is to happen, we have to be able to identify collections of possible outcomes. We call these collections sets. Set A Set is an unordered collection of objects. The objects in a set are called elements of the set. Examples License Plates Poker Hands Students in a Class Even Digits

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Comparing Sets

Comparing two Sets Suppose A and B are sets. Then, Example Let A = {a, b, x}, B = {x, a, b}, and D = {a, b, c, x, y, z}. Which

• f the following comparisons are true?

(a) A = B (b) A ⊂ B (c) A ⊆ D (d) B ⊆ A (e) B ⊆ D (f) A = D

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Comparing Sets

Comparing two Sets Suppose A and B are sets. Then, A = B if A and B have exactly the same elements Example Let A = {a, b, x}, B = {x, a, b}, and D = {a, b, c, x, y, z}. Which

• f the following comparisons are true?

(a) A = B (b) A ⊂ B (c) A ⊆ D (d) B ⊆ A (e) B ⊆ D (f) A = D

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Comparing Sets

Comparing two Sets Suppose A and B are sets. Then, A = B if A and B have exactly the same elements A ⊆ B if every element of A is also an element of B Example Let A = {a, b, x}, B = {x, a, b}, and D = {a, b, c, x, y, z}. Which

• f the following comparisons are true?

(a) A = B (b) A ⊂ B (c) A ⊆ D (d) B ⊆ A (e) B ⊆ D (f) A = D

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Comparing Sets

Comparing two Sets Suppose A and B are sets. Then, A = B if A and B have exactly the same elements A ⊆ B if every element of A is also an element of B A ⊂ B if A ⊆ B and A = B Example Let A = {a, b, x}, B = {x, a, b}, and D = {a, b, c, x, y, z}. Which

• f the following comparisons are true?

(a) A = B (b) A ⊂ B (c) A ⊆ D (d) B ⊆ A (e) B ⊆ D (f) A = D

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Comparing Sets

Comparing two Sets Suppose A and B are sets. Then, A = B if A and B have exactly the same elements A ⊆ B if every element of A is also an element of B A ⊂ B if A ⊆ B and A = B Example Let A = {a, b, x}, B = {x, a, b}, and D = {a, b, c, x, y, z}. Which

• f the following comparisons are true?

(a) A = B (b) A ⊂ B (c) A ⊆ D (d) B ⊆ A (e) B ⊆ D (f) A = D

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Outline

1

Introduction to Sets

2

Combining Sets

3

Visualizing Sets

4

Conclusion

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Union of Two Sets

Set Union If A and B are sets, then the Union of A and B, written A ∪ B, is the set containing all elements in either A or B or both A and B. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d}. Find the following sets.

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Union of Two Sets

Set Union If A and B are sets, then the Union of A and B, written A ∪ B, is the set containing all elements in either A or B or both A and B. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d}. Find the following sets.

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Union of Two Sets

Set Union If A and B are sets, then the Union of A and B, written A ∪ B, is the set containing all elements in either A or B or both A and B. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d}. Find the following sets. (a) A ∪ B

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Union of Two Sets

Set Union If A and B are sets, then the Union of A and B, written A ∪ B, is the set containing all elements in either A or B or both A and B. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d}. Find the following sets. (a) A ∪ B {1, 2, 3, 4, a, c}

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Union of Two Sets

Set Union If A and B are sets, then the Union of A and B, written A ∪ B, is the set containing all elements in either A or B or both A and B. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d}. Find the following sets. (a) A ∪ B {1, 2, 3, 4, a, c} (b) A ∪ C

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Union of Two Sets

Set Union If A and B are sets, then the Union of A and B, written A ∪ B, is the set containing all elements in either A or B or both A and B. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d}. Find the following sets. (a) A ∪ B {1, 2, 3, 4, a, c} (b) A ∪ C {1, 4, a, c, d}

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Union of Two Sets

Set Union If A and B are sets, then the Union of A and B, written A ∪ B, is the set containing all elements in either A or B or both A and B. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d}. Find the following sets. (a) A ∪ B {1, 2, 3, 4, a, c} (b) A ∪ C {1, 4, a, c, d} (c) B ∪ C

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Union of Two Sets

Set Union If A and B are sets, then the Union of A and B, written A ∪ B, is the set containing all elements in either A or B or both A and B. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d}. Find the following sets. (a) A ∪ B {1, 2, 3, 4, a, c} (b) A ∪ C {1, 4, a, c, d} (c) B ∪ C {1, 2, 3, a, d}

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Intersection of Two Sets

Set Intersection If A and B are sets, then the Intersection of A and B, written A ∩ B, is the set containing all elements in both A and B. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d}. Find the following sets.

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Intersection of Two Sets

Set Intersection If A and B are sets, then the Intersection of A and B, written A ∩ B, is the set containing all elements in both A and B. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d}. Find the following sets.

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Intersection of Two Sets

Set Intersection If A and B are sets, then the Intersection of A and B, written A ∩ B, is the set containing all elements in both A and B. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d}. Find the following sets. (a) A ∩ B

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Intersection of Two Sets

Set Intersection If A and B are sets, then the Intersection of A and B, written A ∩ B, is the set containing all elements in both A and B. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d}. Find the following sets. (a) A ∩ B {a}

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Intersection of Two Sets

Set Intersection If A and B are sets, then the Intersection of A and B, written A ∩ B, is the set containing all elements in both A and B. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d}. Find the following sets. (a) A ∩ B {a} (b) A ∩ C

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Intersection of Two Sets

Set Intersection If A and B are sets, then the Intersection of A and B, written A ∩ B, is the set containing all elements in both A and B. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d}. Find the following sets. (a) A ∩ B {a} (b) A ∩ C {1, a}

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Intersection of Two Sets

Set Intersection If A and B are sets, then the Intersection of A and B, written A ∩ B, is the set containing all elements in both A and B. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d}. Find the following sets. (a) A ∩ B {a} (b) A ∩ C {1, a} (c) B ∩ C

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Intersection of Two Sets

Set Intersection If A and B are sets, then the Intersection of A and B, written A ∩ B, is the set containing all elements in both A and B. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d}. Find the following sets. (a) A ∩ B {a} (b) A ∩ C {1, a} (c) B ∩ C {a}

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Universal Set and the Compliment of a Set

Universal Set The set of all possible elements in a given context is called the universal set and denoted by U. Every set is a subset of the universal set. Set Complement The complement of a set A, written A, is the set of all elements of the universal set which are not elements of the set A. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d} be subsets of a universal set U = {1, 2, 3, 4, a, b, c, d}. Find each set. (a) A ∩ B (b) A ∩ A (c) (A ∪ B) ∩ C (d) (A ∩ B) ∩ C (e) A ∩ B (f) A ∪ B

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Universal Set and the Compliment of a Set

Universal Set The set of all possible elements in a given context is called the universal set and denoted by U. Every set is a subset of the universal set. Set Complement The complement of a set A, written A, is the set of all elements of the universal set which are not elements of the set A. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d} be subsets of a universal set U = {1, 2, 3, 4, a, b, c, d}. Find each set. (a) A ∩ B (b) A ∩ A (c) (A ∪ B) ∩ C (d) (A ∩ B) ∩ C (e) A ∩ B (f) A ∪ B

Introduction to Sets Combining Sets Visualizing Sets Conclusion

The Universal Set and the Compliment of a Set

Universal Set The set of all possible elements in a given context is called the universal set and denoted by U. Every set is a subset of the universal set. Set Complement The complement of a set A, written A, is the set of all elements of the universal set which are not elements of the set A. Example Let A = {1, 4, a, c}, B = {2, 3, a}, and C = {1, a, d} be subsets of a universal set U = {1, 2, 3, 4, a, b, c, d}. Find each set. (a) A ∩ B (b) A ∩ A (c) (A ∪ B) ∩ C (d) (A ∩ B) ∩ C (e) A ∩ B (f) A ∪ B

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Outline

1

Introduction to Sets

2

Combining Sets

3

Visualizing Sets

4

Conclusion

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Venn Diagrams

A Venn Diagram is a graphical way to represent one or more sets which are both subsets of a universal set. Universal Set Set A Set B A ∩ B

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Venn Diagrams

A Venn Diagram is a graphical way to represent one or more sets which are both subsets of a universal set. Universal Set Set A Set B A ∩ B U

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Venn Diagrams

A Venn Diagram is a graphical way to represent one or more sets which are both subsets of a universal set. Universal Set Set A Set B A ∩ B U A

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Venn Diagrams

A Venn Diagram is a graphical way to represent one or more sets which are both subsets of a universal set. Universal Set Set A Set B A ∩ B U A B

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Venn Diagrams

A Venn Diagram is a graphical way to represent one or more sets which are both subsets of a universal set. Universal Set Set A Set B A ∩ B U A B

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Placing Elements in a Venn Diagram

Example Let U be the universal set of all possible outcomes when six-sided die is rolled. Let A be the set of even outcomes, B the set of odd

• utcomes, and C the set of outcomes less than or equal to three.

Use Venn Diagrams to find A ∪ C, A ∩ C, A ∩ C, and A ∩ B. A C 4 6 2 1 3 A B 2 4 6 1 3 5

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Placing Elements in a Venn Diagram

Example Let U be the universal set of all possible outcomes when six-sided die is rolled. Let A be the set of even outcomes, B the set of odd

• utcomes, and C the set of outcomes less than or equal to three.

Use Venn Diagrams to find A ∪ C, A ∩ C, A ∩ C, and A ∩ B. A C 4 6 2 1 3 A B 2 4 6 1 3 5

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Placing Elements in a Venn Diagram

Example Let U be the universal set of all possible outcomes when six-sided die is rolled. Let A be the set of even outcomes, B the set of odd

• utcomes, and C the set of outcomes less than or equal to three.

Use Venn Diagrams to find A ∪ C, A ∩ C, A ∩ C, and A ∩ B. A C 4 6 2 1 3 A B 2 4 6 1 3 5

Introduction to Sets Combining Sets Visualizing Sets Conclusion

DeMorgan’s Laws

The last example is one of two “Laws” which are often useful.

DeMorgan #1

A B A ∪ B = A ∩ B

DeMorgan #2

A B A ∩ B = A ∪ B

Introduction to Sets Combining Sets Visualizing Sets Conclusion

DeMorgan’s Laws

The last example is one of two “Laws” which are often useful.

DeMorgan #1

A B A ∪ B = A ∩ B

DeMorgan #2

A B A ∩ B = A ∪ B

Introduction to Sets Combining Sets Visualizing Sets Conclusion

DeMorgan’s Laws

The last example is one of two “Laws” which are often useful.

DeMorgan #1

A B A ∪ B = A ∩ B

DeMorgan #2

A B A ∩ B = A ∪ B

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Outline

1

Introduction to Sets

2

Combining Sets

3

Visualizing Sets

4

Conclusion

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Important Concepts

Things to Remember from Section 6-1

1 The several ways to write sets: list, set-builder, description 2 Comparing sets: equal sets, subsets, disjoint sets 3 Combining sets: unions and intersections 4 Universal sets and set complements 5 Visualizing sets using Venn Diagrams

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Important Concepts

Things to Remember from Section 6-1

1 The several ways to write sets: list, set-builder, description 2 Comparing sets: equal sets, subsets, disjoint sets 3 Combining sets: unions and intersections 4 Universal sets and set complements 5 Visualizing sets using Venn Diagrams

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Important Concepts

Things to Remember from Section 6-1

1 The several ways to write sets: list, set-builder, description 2 Comparing sets: equal sets, subsets, disjoint sets 3 Combining sets: unions and intersections 4 Universal sets and set complements 5 Visualizing sets using Venn Diagrams

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Important Concepts

Things to Remember from Section 6-1

1 The several ways to write sets: list, set-builder, description 2 Comparing sets: equal sets, subsets, disjoint sets 3 Combining sets: unions and intersections 4 Universal sets and set complements 5 Visualizing sets using Venn Diagrams

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Important Concepts

Things to Remember from Section 6-1

1 The several ways to write sets: list, set-builder, description 2 Comparing sets: equal sets, subsets, disjoint sets 3 Combining sets: unions and intersections 4 Universal sets and set complements 5 Visualizing sets using Venn Diagrams

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Important Concepts

Things to Remember from Section 6-1

1 The several ways to write sets: list, set-builder, description 2 Comparing sets: equal sets, subsets, disjoint sets 3 Combining sets: unions and intersections 4 Universal sets and set complements 5 Visualizing sets using Venn Diagrams

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Next Time. . .

As we mentioned earlier, probability is about comparing the sizes

• f sets of different outcomes. In the next section we start counting

elements in sets and using Venn Diagrams to make this easier. For next time Read section 6-2 (pp 326-329) Review your syllabus Do problem set 6-1 A

Introduction to Sets Combining Sets Visualizing Sets Conclusion

Next Time. . .

As we mentioned earlier, probability is about comparing the sizes

• f sets of different outcomes. In the next section we start counting

elements in sets and using Venn Diagrams to make this easier. For next time Read section 6-2 (pp 326-329) Review your syllabus Do problem set 6-1 A