Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A B= {3}.

Transcription

1 Section 1.3: Intersection and Union of Two Sets Exploring the Different Regions of a Venn Diagram There are 6 different set notations that you must become familiar with. 1. The intersection is the set of elements that are common to two or more sets (i.e. they are in both sets). In set notation, A B, denotes the intersection of sets A and B. Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A B= {3}. The intersection of two or more set is represented by the OVERLAP on a Venn diagram. It is indicated by the word "and". 1

2 2. The union is the set of all elements in two or more sets (i.e. all in elements in at least one of the sets). In set notation, A B, denotes the union of sets A and B. Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A B = {1, 2, 3, 4, 5}. It is represented by the ENTIRE region of these sets on a Venn diagram (includes everything in BOTH circles). It is indicated by the word "or". Disjoint Sets Sets with Common Elements A B A B n(a B) = n(a) + n(b) n(a B) = n(a) + n(b) - n(a B) This is called the Principle of Inclusion and Exclusion. We subtract the elements in the intersection so they are not counted twice. 2

3 3. Set A minus Set B, is the set of elements found in set A but excluding the ones that are also in set B. In other words, A\B means elements found ONLY in set A. Notation: A\B If A = {1, 2, 3} and B = {3, 4, 5}, then A\B = {1, 2} A 1,2 3 4, 5 B Formula: n(a\b) = n(a) - n(a B) n(a\b) = n(a B) - n(b) 3

4 HOMEWORK Example: The universal set is the set of all integers from -3 to +3. Set A is the set of non-negative integers and set B is the set of integers divisible by 2. Complete the following table by draw the Venn diagram and writing the elements that correspond to the notation. 1 Set Notation A B (A union B) Meaning Venn Diagram Answer any element that is in either of the sets 2 3 A B (A intersect B) A\B set A minus set B only elements that are in both A and B elements found in set A but excluding the ones that are also in set B 4

5 n(a B) = n(a) + n(b) - n(a B) n(a\b) = n(a) - n(a B) n(a\b) = n(a B) - n(b) 5

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7 Example 1. Consider the following samples shown below: (a) The diagrams below represent the activities chosen by youth club members. They can choose to play tennis (T), baseball (B) or swimming (S). Decide which diagram has the shading which represents the following descriptions: (i) those who play all three sports (ii) those who play tennis and baseball, but not swimming (iii) those who play only tennis 7

8 (b) The diagrams below represent a class of children. G is the set of girls and F is the set of children who like fencing. Decide which diagram has the shading which represents: (i) girls who like fencing (ii) girls who dislike fencing (iii) boys who like fencing (iv) boys who dislike fencing 8

9 2. Jamaal surveyed 34 people at his gym. He learned that 16 people do weight training three times a week, 21 people do cardio training three times a week, and 6 people train fewer than 3 times a week. A) How many people do cardio and weight training 3 times a week? Use a Venn diagram and the Principle of Inclusion and Exclusion to answer the question. B) How many people do only weight training? C) How many people do only cardio training? 9

10 3. Morgan surveyed the 30 students in her mathematics class about their eating habits. 18 students eat breakfast 5 of the 18 students also eat a healthy lunch 3 students do not eat breakfast and do not eat a healthy lunch. How many students eat a healthy lunch? Use a Venn diagram and the Principle of Inclusion and Exclusion to answer the question. 10

11 4. Tyler asked 55 people if they like Criminal Minds or Chicago Fire. - 8 people didn't like either show - 20 people liked Criminal Minds - 38 people liked Chicago Fire Determine how many people liked both shows, how many only liked Criminal Minds, and how many people liked only Chicago Fire. Use a Venn diagram and the Principle of Inclusion and Exclusion. 11

12 4. Jason asked 100 people if they liked Pepsi or 7-UP people didn't like either drink - 18 liked both Pepsi and 7-UP - 25 people liked only 7-UP Determine how many people liked only Pepsi. 12

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