MATH1215: Mathematical Thinking Sec. 08 Spring Worksheet 3: Solution

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1 N. Name: Some useful notations: MTH121: Mathematical Thinking Sec. 08 Spring 2011 Worksheet 3: Solution (i) Difference of sets. Given and, define the set The corresponding region in the Venn diagram is: \ = {x x x } = (ii) The power set of a set is the set of all subsets of. This includes the subsets formed from all the members of and the empty set. If a finite set has cardinality k, then the power set of has cardinality 2 k. The power set can be written as P(). If =, that is k = 0, then there is only one subset, the empty set, that is itself: P() = { } P() = 1 If = 1 and = {a}, then we have the empty set and the subset {a} =. So If = 2, = {a, b}, we have: If = 3, = {a, b, c}, we have: P() = {, {a} } P() = 2 P() = {, {a}, {b}, {a, b} } P() = 4 P() = {, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} } P() = 8 (iii) We can represent the power set with an ordered diagram. For example, if = {a, b, c}, then P() is: 3 2 {a, b} {a, c} {b, c} 1 {a} {b} {c} 0 where a subset P on a level is connected with a subset Q in the upper level if and only if P Q. So, for example, since {a} {a, b}, then {a} on level 1 is connected with {a, b} on level 2. Then number of the level is the cardinality of the subsets on that level. 1

2 Exercises: 1) onsider the universal set and the subsets = {1, 2, 3, 4,, 6, 7, 8} E = {1, 3,, 6} F = {2, 4,, 7} (a) Describe the following subsets: a) E F = {2, 4, 7, 8} {2, 4,, 7} = {2, 4,, 7, 8} b) E E = c) E F = {1, 3,, 6, 8} d) E F = {1, 3,, 6} {1, 3, 6, 8} = {1, 3, 6} e) F F = (b) Draw the Venn diagram for, E, F and write the number of elements for each region: 1 E F ) Let be the universal set and,, subsets. ssume = 100, = 39, = 11, = 43 and = 74, = 41, = 73, = 49. Remember that the union rule is given by: and, in the case of three sets, = + = (a) Draw the Venn diagram for,,, and write the number of elements for each region. ns. The first information we can get is about the gray region (see below): ( ) = = = 26 Then, using the union rule, we get the following informations: = + = = 9 = + = = = + = = 9 In particular, we can determine the brown region: = = = 4 Now, using the fact that = 4, we can say that: Violet region: ( )\ = = 9 4 = Orange region: ( )\ = = 4 = 1 Green region: ( )\ = = 9 4 = p to now, we have the following situation: 2

3 Then, it is easy to find: lue region: \( ) = = 2 Red region: \( ) = = 1 Yellow region: \( ) = Finally, we have: (b) Shade the following regions and write their cardinality: (a) = = = 61 (b) = 4 (c) ( ) = ( ) ( ) = 43 + = 48 3

4 (d) ( ) = = 31 (e) ( ) = [( ) ] = [( ) ( )] = = 90 3) Decide if the following statements are true or false, where is the universal set and, are sets: (a) If, then <. F (b) = T (c) If a, then a ( ) T (d) The empty set has no subset. F (e) set of 4 elements has 8 subsets. F (f) If =, then P() = 32. T (g) + = + T 4) (a) Write the union rule for sets, ns. = + (b) ssume and are disjoint, with = 18 and = 4. Find. ns. Since, are disjoint, = 0, then = + = = 22. (c) Let = 8, = 10, =, = 12. Find. ns. sing union rule, = + = = 9 then, since, are disjoint and =, = + = = 19 4

5 ) n Italian restaurant surveyed 120 customers to find out what topping they ordered for their pizzas among olives (O), artichokes (), and mushrooms (M). It was found that: a. 14 ordered all three toppings; b. 13 ordered no toppings; c. 26 ordered artichokes and olives; d. 18 ordered artichokes and mushrooms; e. 30 ordered olives and mushrooms; f. 67 ordered artichokes; g. 90 ordered anchovies or mushrooms. (a) se the information to fill in the number of elements for each region in the following Venn diagram. ns. The given information correspond to: a. O M = 14 (brown region) b. ( O M) = 13 (gray region) c. O = 26 d. M = 18 e. O M = 30 f. = 67 g. M = 90 sing (a) we get the data for colors violet,green, orange: ( O)\M = O O M = = 12 ( M)\O = M O M = = 4 (M O)\ = M O O M = = 16 p to now we have the following information: 13 M O Now, using (a), we get: O M = ( O M) = = 107 and by (f) we get (blue region): \(M O) = ( ) = = 37 and by (g), we have (yellow region): M\( O) = M ( ) = = 7 and finally (red region): O\( M) = O M M = = 17 So, the Venn diagram is:

6 7 M O 17 (b) How many customers did not order olives? ns. The request corresponds to find the cardinality of O, that is 7 M O 17 and the cardinality is given by: O = O = = 61 6) local town has its own electric company which also provides both digital cable TV () and highspeed internet access (I). The electric company has 200 customers. 70 customers subscribe to digital cable TV, 0 customers subscribe to high-speed internet access, and 110 customers subscribe to neither. How many customers subscribe to both digital cable TV and high-speed internet access? ns. In order to answer the question we have to compute I, knowing the value for = 200, = 70, I = 0, (I ) = 110. Now, therefore, using union rule, we get: (I ) = I = I = = 90 I = I + I = = 30 7) Some colleges were surveyed to determine sports teams. 34 had Football teams (F), 39 had asketball teams (), 1 had Volleyball (V) or asketball teams, 9 had Football and Volleyball teams, 11 had Volleyball and asketball teams, 13 had Football and asketball teams, 6 had all three and 11 colleges had none of these 3 teams. (a) Fill in the given Venn Diagram. ns. List all the information we have: F = 34 = 39 V = 1 F V = 9 V = 11 F = 13 (brown) F V = 6 (gray) (F V ) = 11 6

7 These are the information we can recover from the above: (green) (F V )\ = F V F V = 9 6 = 3 (orange) ( V )\F = V F V = 11 6 = (violet) (F )\V = F F V = 13 6 = 7 that fit in the Venn diagram: 11 F V Then V = V + V = = 23 (yellow) V \(F ) = V ( ) = = 9 (red) \(F V ) = V (23 + 7) = 1 30 = 21 (blue) F \( V ) = F ( ) = = 18 and the complete Venn diagram is: 9 V F 7 21 (b) How many colleges were surveyed? ns. = = 80 (c) How many had only one of these teams? ns. The question concerns the area 9 V F 7 21 that is formed by = 38 elements. (d) How many had Football or asketball teams? ns. F = F + F = = 60 (e) How many had asketball teams but not Football teams? 7

8 ns. \F = F = = 26 8) group of 10 people go to Haymarket on Saturday morning between 10am-11am. 90 bought apples (); 0 bought bananas (); 70 bought cherries (); 1 bought apples and cherries; 12 bought bananas and cherries; 10 bought all three products; 3 bought none of them. Determine how many people bought apples and bananas, using the Venn diagram. ns. List all the information we have: = 10 = 90 = 0 = 70 = 1 = 12 = 10 (brown) ( ) = 3 (gray) Here there are the first information we can deduce: = ( ) = 10 3 = 147 (green) ( )\ = = 1 10 = (orange) ( )\ = = = 2 (yellow) \( ) = ( ) = = 3 We end up with the following diagram: Now, call x the cardinality of the violet region, that is, x = ( )\. We know that: (blue) \( ) = x = 90 1 x = 7 x (red) \( ) = x = 0 12 x = 38 x So everything is reduced to x. ut, we know that the = 10, that is, Finally then 10 = (7 x) + x + (38 x) = 186 x = x = = 36 (violet) ( )\ = 36 8

9 (blue) \( ) = 7 36 = 39 (red) \( ) = = 2 and the complete Venn diagram is: The number of people that bought apples and bananas is given by = = 46 9) Represents all the subset of = {1, 2, 3, 4} using the diagram described at the beginning of this worksheet. ns. We will have 2 4 = 16 subsets, ordered on five levels. The resulting diagram is: 4 3 {1, 2, 3} {1, 2, 4} {1, 3, 4} {2, 3, 4} 2 {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} 1 {1} {2} {3} {4} 0 9