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2.3 – Set Operations and Cartesian Products

2.3 – Set Operations and Cartesian Products. Intersection of Sets : The intersection of sets A and B is the set of elements common to both A and B. A  B = {x | x  A and x  B}. {1, 2, 5, 9, 13}  {2, 4, 6, 9}. {2, 9}. {a, c, d, g}  {l, m, n, o}. .

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2.3 – Set Operations and Cartesian Products

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  1. 2.3 – Set Operations and Cartesian Products Intersection of Sets: The intersection of sets A and B is the set of elements common to both A and B. • A  B = {x | x  A and x  B} • {1, 2, 5, 9, 13}  {2, 4, 6, 9} • {2, 9} • {a, c, d, g}  {l, m, n, o} •  • {4, 6, 7, 19, 23}  {7, 8, 19, 20, 23, 24} • {7, 19, 23}

  2. 2.3 – Set Operations and Cartesian Products Union of Sets: The union of sets A and B is the set of all elements belonging to each set. • A  B = {x | x  A or x  B} • {1, 2, 5, 9, 13}  {2, 4, 6, 9} • {1, 2, 4, 5, 6, 9, 13} • {a, c, d, g}  {l, m, n, o} • {a, c, d, g, l, m, n, o} • {4, 6, 7, 19, 23}  {7, 8, 19, 20, 23, 24} • {4, 6, 7, 8, 19, 20, 23, 24}

  3. 2.3 – Set Operations and Cartesian Products Find each set. • U = {1, 2, 3, 4, 5, 6, 9} • A = {1, 2, 3, 4} B = {2, 4, 6} C = {1, 3, 6, 9} • A  B • {1, 2, 3, 4, 6} • A B • A= {5, 6, 9} • {6} • B C • B= {1, 3, 5, 9)} • C= {2, 4, 5} • {1, 2, 3, 4, 5, 9} • B B • 

  4. 2.3 – Set Operations and Cartesian Products Find each set. • U = {1, 2, 3, 4, 5, 6, 9} • A = {1, 2, 3, 4} B = {2, 4, 6} C = {1, 3, 6, 9} • A= {5, 6, 9} • B= {1, 3, 5, 9)} • C= {2, 4, 5} • (A C)  B • A C • {2, 4, 5, 6, 9} • {2, 4, 5, 6, 9}  B • {5, 9}

  5. 2.3 – Set Operations and Cartesian Products Difference of Sets: The difference of sets A and B is the set of all elements belonging set A and not to set B. • A – B = {x | x  A and x  B} • U = {1, 2, 3, 4, 5, 6, 7} • A = {1, 2, 3, 4, 5, 6} B = {2, 3, 6} C = {3, 5, 7} • A= {7} • B= {1, 4, 5, 7} • C= {1, 2, 4, 6} Find each set. • A – B • {1, 4, 5} • B – A •  • Note: A – B  B – A • (A – B)  C • {1, 2, 4, 5, 6, }

  6. 2.3 – Set Operations and Cartesian Products Ordered Pairs: in the ordered pair (a, b), a is the first component and b is the second component. In general, (a, b)  (b, a) Determine whether each statement is true or false. • (3, 4) = (5 – 2, 1 + 3) • True • {3, 4}  {4, 3} • False • (4, 7) = (7, 4) • False

  7. 2.3 – Set Operations and Cartesian Products Cartesian Product of Sets: Given sets A and B, the Cartesian product represents the set of all ordered pairs from the elements of both sets. A  B = {(a, b) | a  A and b  B} Find each set. • A = {1, 5, 9} • B = {6,7} • A  B • { • (1, 6), • (5, 6), • (1, 7), • (5, 7), • (9, 6), • (9, 7) • } • B  A • { • (6, 1), • (6, 9), • (6, 5), • (7, 1), • (7, 5), • (7, 9) • }

  8. 2.3 – Venn Diagrams and Subsets Shading Venn Diagrams: • A  B A B U A B A B U U

  9. 2.3 – Venn Diagrams and Subsets Shading Venn Diagrams: • A  B A B U A A B B U U

  10. 2.3 – Venn Diagrams and Subsets Shading Venn Diagrams: • A B A B U A A B A B U U • A B in yellow

  11. 2.3 – Venn Diagrams and Subsets Locating Elements in a Venn Diagram • U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} • A = {2, 3, 4, 5, 6} B = {4, 6, 8} • Start with A  B 7 1 • Fill in each subset of U. A B 4 2 3 8 • Fill in remaining elements of U. 6 5 U 9 10

  12. 2.3 – Venn Diagrams and Subsets Shade a Venn diagram for the given statement. • (A  B)  C Work with the parentheses. (A  B) • A • B • C • U

  13. 2.3 – Venn Diagrams and Subsets Shade a Venn diagram for the given statement. • (A  B)  C Work with the parentheses. (A  B) Work with the remaining part of the statement. • A • B (A  B)  C • C • U

  14. 2.3 – Venn Diagrams and Subsets Shade a Venn diagram for the given statement. • (A  B)  C Work with the parentheses. (A  B) Work with the remaining part of the statement. • A • B (A  B)  C • C • U

  15. 2.4 –Surveys and Cardinal Numbers Surveys and Venn Diagrams • Financial Aid Survey of a Small College (100 sophomores). • 49 received Government grants • 55 received Private scholarships • 43 received College aid G P • 23 received Gov. grants & Pri. scholar. 16 15 12 • 18 received Gov. grants & College aid 8 • 28 received Pri. scholar. & College aid 20 10 • 8 received funds from all three 5 (PC) – (GPC) 28 – 8 = 20 43 – (10 + 8 +20) = 5 C U 14 (GC) – (GPC) 18 – 8 = 10 55 – (15 + 8 + 20) = 12 (GP) – (GPC) 23 – 8 = 15 49 – (15 + 8 + 10) = 16 100 – (16+15 + 8 + 10+12+20+5) = 14

  16. 2.4 –Surveys and Cardinal Numbers Cardinal Number Formula for a Region For any two sets A and B, Find n(A) if n(AB) = 78, n(AB) = 21, and n(B) = 36. n(AB) = n(A) + n(B ) – n(AB) 78 = n(A) + 36 – 21 78 = n(A) + 15 63 = n(A)

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