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 Introduction to Sets 1 Combining Sets 2 Visualizing Sets 3 Conclusion 4
Introduction to Sets Combining Sets Visualizing Sets Conclusion Outline Introduction to Sets 1 Combining Sets 2 Visualizing Sets 3 Conclusion 4
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 of 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 of 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 of 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 of 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 of 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 Introduction to Sets 1 Combining Sets 2 Visualizing Sets 3 Conclusion 4
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
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