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2.1 – Symbols and Terminology

2.1 – Symbols and Terminology. Definitions:. Set: A collection of objects. . Elements: The objects that belong to the set. . Set Designations (3 types):. Word Descriptions:. The set of even counting numbers less than ten. Listing method:. {2, 4, 6, 8}. Set Builder Notation:.

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2.1 – Symbols and Terminology

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  1. 2.1 – Symbols and Terminology Definitions: • Set: A collection of objects. • Elements: The objects that belong to the set. Set Designations (3 types): • Word Descriptions: • The set of even counting numbers less than ten. • Listing method: • {2, 4, 6, 8} • Set Builder Notation: • {x | x is an even counting number less than 10}

  2. 2.1 – Symbols and Terminology Definitions: • Empty Set: A set that contains no elements. It is also known as the Null Set. The symbol is  • List all the elements of the following sets. • The set of counting numbers between six and thirteen. • {7, 8, 9, 10, 11, 12} • {5, 6, 7,…., 13} • {5, 6, 7, 8, 9, 10, 11, 12, 13} • {x | x is a counting number between 6 and 7} { } •  • Null set • Empty set

  3. 2.1 – Symbols and Terminology Symbols: • ∈: Used to replace the words “is an element of.” • ∉: Used to replace the words “is not an element of.” True or False: • 3∈ {1, 2, 5, 9, 13} • False • 0 ∈ {0, 1, 2, 3} • True • True • -5 ∉ {5, 10, 15, , }

  4. 2.1 – Symbols and Terminology Sets of Numbers and Cardinality Cardinal Number or Cardinality: The number of distinct elements in a set. Notation • n(A): n of A; represents the cardinal number of a set. • K= {2, 4, 8, 16} • n(K) = 4 • ∅ • n(∅) = 0 • R = {1, 2, 3, 2, 4, 5} • n(R) = 5 • P = {∅} • n(P) = 1

  5. 2.1 – Symbols and Terminology Finite and Infinite Sets Finite set: The number of elements in a set are countable. Infinite set: The number of elements in a set are not countable • {2, 4, 8, 16} • Countable = Finite set • Not countable = Infinite set • {1, 2, 3, …}

  6. 2.1 – Symbols and Terminology Equality of Sets Set A is equal to set B if the following conditions are met: 1. Every element of A is an element of B. 2. Every element of B is an element of A. • Are the following sets equal? • {–4, 3, 2, 5} and {–4, 0, 3, 2, 5} • Not equal • {3} = {x | x is a counting number between 2 and 5} • Not equal • {11, 12, 13,…} = {x | x is a natural number greater than 10} • Equal

  7. 2.2 – Venn Diagrams and Subsets Definitions: • Universal set: the set that contains every object of interest in the universe. • Complement of a Set: A set of objects of the universal set that are not an element of a set inside the universal set. Notation: A • Venn Diagram: A rectangle represents the universal set and circles represent sets of interest within the universal set A A U

  8. 2.2 – Venn Diagrams and Subsets Definitions: • Subset of a Set: Set A is a Subset of B if every element of A is an element of B. Notation: AB • Subset or not?  • {3, 4, 5, 6} {3, 4, 5, 6, 8}  • {1, 2, 6} {2, 4, 6, 8}  • {5, 6, 7, 8} {5, 6, 7, 8} • BB • Note: Every set is a subset of itself.

  9. 2.2 – Venn Diagrams and Subsets Definitions: • Set Equality: Given A and B are sets, then A = B if AB and BA. = • {1, 2, 6} {1, 2, 6}  • {5, 6, 7, 8} {5, 6, 7, 8, 9}

  10. 2.2 – Venn Diagrams and Subsets Definitions: • Proper Subset of a Set: Set A is a proper subset of Set B if AB and A  B. Notation AB • What makes the following statements true? • , , or both both • {3, 4, 5, 6} {3, 4, 5, 6, 8} both • {1, 2, 6} {1, 2, 4, 6, 8}  • {5, 6, 7, 8} {5, 6, 7, 8} • The empty set () is a subset and a proper subset of every set except itself.

  11. 2.2 – Venn Diagrams and Subsets Number of Subsets • The number of subsets of a set with n elements is: 2n • Number of Proper Subsets • The number of proper subsets of a set with n elements is: 2n – 1 • List the subsets and proper subsets • {1, 2} • {1} • 22 = 4 • {2} • Subsets: • {1,2} •  • Proper subsets: • 22 – 1= 3 • {1} • {2} • 

  12. 2.2 – Venn Diagrams and Subsets • List the subsets and proper subsets • {a, b, c} • {a} • {b} • Subsets: • {c} • {a, b} • {a, c} • {b, c} • 23 = 8 • {a, b, c} •  • Proper subsets: • {a} • {b} • {c} • {a, b} • {a, c} • {b, c} • 23 – 1 = 7 • 

  13. 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}

  14. 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}

  15. 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 • 

  16. 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}

  17. 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, }

  18. 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

  19. 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) • }

  20. 2.3 – Venn Diagrams and Subsets Shading Venn Diagrams: • A  B A B U A B A B U U

  21. 2.3 – Venn Diagrams and Subsets Shading Venn Diagrams: • A  B A B U A A B B U U

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

  27. 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

  28. 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)

  29. 9.1 – Points, Line, Planes and Angles Definitions: A point has no magnitude and no size. A line has no thickness and no width and it extends indefinitely in two directions. A plane is a flat surface that extends infinitely. m A E D

  30. 9.1 – Points, Line, Planes and Angles Definitions: A point divides a line into two half-lines, one on each side of the point. A ray is a half-line including an initial point. A line segment includes two endpoints. N E D G F

  31. 9.1 – Points, Line, Planes and Angles Summary: Line AB or BA AB BA B A Half-line AB AB A B Half-line BA BA A B Ray AB AB A B Ray BA BA A B Segment AB or Segment BA BA A B AB

  32. 9.1 – Points, Line, Planes and Angles Definitions: Parallel lines lie in the same plane and never meet. Two distinct intersecting lines meet at a point. Skew lines do not lie in the same plane and do not meet. Intersecting Skew Parallel

  33. 9.1 – Points, Line, Planes and Angles Definitions: Parallel planes never meet. Two distinct intersecting planes meet and form a straight line. Parallel Intersecting

  34. 9.1 – Points, Line, Planes and Angles Definitions: An angle is the union of two rays that have a common endpoint. A Side 1 Vertex B Side C An angle can be named using the following methods: – with the letter marking its vertex, B – with the number identifying the angle, 1 – with three letters, ABC. 1) the first letter names a point one side; 2) the second names the vertex; 3) the third names a point on the other side.

  35. 9.1 – Points, Line, Planes and Angles Angles are measured by the amount of rotation in degrees. Classification of an angle is based on the degree measure. Between 0° and 90° Acute Angle 90° Right Angle Greater than 90° but less than 180° Obtuse Angle Straight Angle 180°

  36. 9.1 – Points, Line, Planes and Angles When two lines intersect to form right angles they are called perpendicular. Vertical angles are formed when two lines intersect. A D B E C ABC and DBE are one pair of vertical angles. DBA and EBC are the other pair of vertical angles. Vertical angles have equal measures.

  37. 9.1 – Points, Line, Planes and Angles Complementary Angles and Supplementary Angles If the sum of the measures of two acute angles is 90°, the angles are said to be complementary. Each is called the complement of the other. Example: 50° and 40° are complementary angles. If the sum of the measures of two angles is 180°, the angles are said to be supplementary. Each is called the supplement of the other. Example: 50° and 130° are supplementary angles

  38. 9.1 – Points, Line, Planes and Angles Find the measure of each marked angle below. (3x + 10)° (5x – 10)° Vertical angels are equal. 3x + 10 = 5x – 10 2x = 20 x = 10 Each angle is 3(10) + 10 = 40°.

  39. 9.1 – Points, Line, Planes and Angles Find the measure of each marked angle below. (2x + 45)° (x – 15)° Supplementary angles. 2x + 45 + x – 15 = 180 3x + 30 = 180 3x = 150 x = 50 2(50) + 45 = 145 50 – 15 = 35 35° + 145° = 180

  40. 9.1 – Points, Line, Planes and Angles 1 2 Parallel Lines cut by a Transversal line create 8 angles 3 4 5 6 7 8 Alternate interior angles 5 4 Angle measures are equal. (also 3 and 6) 1 Alternate exterior angles Angle measures are equal. 8 (also 2 and 7)

  41. 9.1 – Points, Line, Planes and Angles 1 2 3 4 5 6 7 8 Same Side Interior angles 4 Angle measures add to 180°. 6 (also 3 and 5) 2 Corresponding angles 6 Angle measures are equal. (also 1 and 5, 3 and 7, 4 and 8)

  42. 9.1 – Points, Line, Planes and Angles Find the measure of each marked angle below. (3x – 80)° (x + 70)° Alternate interior angles. x + 70 = x + 70 = 3x – 80 75 + 70 = 2x = 150 145° x = 75

  43. 9.1 – Points, Line, Planes and Angles Find the measure of each marked angle below. (4x – 45)° (2x – 21)° Same Side Interior angles. 4(41) – 45 4x – 45 + 2x – 21 = 180 2(41) – 21 164 – 45 6x – 66 = 180 82 – 21 119° 61° 6x = 246 x = 41 180 – 119 = 61°

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