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Sets and Counting

6. Sets and Counting. Sets and Set Operations The Number of Elements in a Finite Set The Multiplication Principle Permutations and Combinations. Set: A set is a collection of objects/elements. Ex. A = { w , a , r , d }. Order of elements doesn’t matter, no duplicates.

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Sets and Counting

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  1. 6 Sets and Counting • Sets and Set Operations • The Number of Elements in a Finite Set • The Multiplication Principle • Permutations and Combinations

  2. Set: A set is a collection of objects/elements. Ex.A = {w, a, r, d} Order of elements doesn’t matter, no duplicates. Sets are often named with capital letters. Notation: w is an element of set A is written Set-builder notation: rule describes the definite property (properties) an object x must satisfy to be part of the set. Ex.B = {x | x is an even integer} Read: “x such that x is an even integer”

  3. Set Equality: Two sets A and B are equal, written A = B, if and only if they have exactly the same elements. Ex.A = {w, a, r, d}; B = {d, r, a, w} Every element in A is in B and every element in B is in A. Subset: If every element of a set A is also an element of a set B then A is a subset of B, written Ex.A = {r, d}; B = {r, a, w, d, e, t} Every element in A is also in B

  4. Empty Set: The set that contains no elements is called the empty set and is denoted Note: The empty set is a subset of every set Universal Set: The set of all elements of interest in a particular discussion is called the universal set and is denoted U.

  5. Set Operations Set Union: Let A and B be sets. The union of A and B, written is the set of all elements that belong to either A or B. Set Intersection: Let A and B be sets. The intersection of A and B, written is the set of all elements that are common to A and B.

  6. Ex. Given the sets: Combine the sets Overlap of the sets

  7. Venn Diagrams – visual representation of sets Rectangle = Universal Set U A B Sets are represented by circles

  8. Venn Diagrams U A B C U A B C

  9. Complement of a Set: If U is a universal set and A is a subset of U, then the set of all elements in U that are not in A is called the complement of A, written AC. Set Complementation

  10. Set Operations Commutative Laws Associative Laws Distributive Laws

  11. De Morgan’s Laws Let A and B be sets, then

  12. Ex. Given the sets: Elements not in A. Elements in A and not in B.

  13. Venn Diagrams U A U A B

  14. The Number of Elements in a Set The number of elements in a set A is denoted n(A). Ex. Given n(A) = 4 Since the union doesn’t count a and h twice Notice This leads to Overlap is subtracted

  15. Venn Diagram – number of elements U 22 B A 5 12 31 We can see that So which leads to

  16. Survey In a survey of 100 people at a carnival: 40 like cotton candy 30 like popcorn 45 like lemonade 15 like lemonade and popcorn 10 like cotton candy and lemonade 12 like cotton candy and popcorn 5 like all three How many people don’t like lemonade, popcorn, or cotton candy? How many people only like popcorn?

  17. Survey set C: cotton candy, set P: popcorn, set L: lemonade U P L 10 8 25 5 7 5 n(C) = 40, n(P) = 30, n(L) = 45 23 C 17 n(U) = 100 8 people only like popcorn. 17 people don’t like lemonade, popcorn, or cotton candy.

  18. The Multiplication Principle If there are m ways of performing a task T1 and n ways of performing a task T2, then there are mn ways of performing task T1 followed by task T2. Note: This generalizes to the multiplication principle involving more than two tasks. Ex. A man has a choice of 8 shirts and 3 different pants. How many outfits can the man wear? Considering shirts as task 1 and pants as task 2, we have 8(3) = 24 outfits

  19. Ex. How many outcomes are possible for a game that consists of rolling a die followed by flipping a fair coin? 6 possibilities 2 possibilities Total of 6(2) = 12 outcomes

  20. Ex. An employee ID for a particular company consists of the employee’s first initial, last initial, and last four digits of his/her social security number. How many possible ID’s are there? Each Initial has 26 possibilities (A–Z) and each digit has 10 possibilities (0–9) 26(26)(10)(10)(10)(10) = 6760000 6760000 different IDs possible

  21. Permutations A permutation of a set of objects is an arrangement of these objects in a definite order. Combinations A combination is a selection of r objects from a set of n objects where order is not important

  22. n–Factorial For any natural number n, Ex. 5! = 5(4)(3)(2)(1) = 120 Ex. This notation allows us to write expressions associated with permutations and combinations in a compact form.

  23. Permutations of n Distinct Objects The number of permutations of n distinct objects taken r at a time is given by Ex.

  24. Ex. A boy has 4 beads – red, white, blue, and yellow. How different ways can three of the beads be strung together in a row? This is a permutation since the beads will be in a row (order). 24 different ways total number selected

  25. Permutations of n Objects, Not all Distinct Given n objects with n1 (non-distinct) of type 1, n2 (non-distinct) of type 2,…, nr (non-distinct) of type r where n = n1 + n2 + … + nr then the number of permutations of these n objects taken n at a time is given by

  26. Ex. How many distinguishable arrangements are there of the letters of the word initializing? There are 12 letters n appears 2 times i appears 5 times

  27. Combinations of n Objects The number of combinations of n distinct objects taken r at a time is given by Ex. Find C(9, 6). = 84

  28. Ex 1. A boy has 4 beads – red, white, blue, and yellow. How different ways can three of the beads be chosen to trade away? This is a combination since they are chosen without regard to order. 4 different ways total number selected

  29. Ex 2. A space shuttle crew consists of a shuttle commander, a pilot, three engineers, a scientist, and a civilian. The shuttle commander and pilot are to be chosen from 8 candidates, the three engineers from 12 candidates, the scientist from 5 candidates, and the civilian from 2 candidates. How many such space shuttle crews can be formed? There are P(8, 2) ways of picking the shuttle commander and pilot (the order is important here), C(12,3) ways of picking the engineers (the order is not important here), C(5, 1) ways of picking the scientist, and C(2, 1) ways of picking the civilian. . . .

  30. By the multiplication principle:

  31. Assignment for Chapter 6 • §6.1 4,8,14,20,24,28,32,36,46,52,60,72 • §6.2 2, 8,18,20,28,32,36,40 • §6.3 2,4,6,8,14,18,22,26,28 • §6.4 1-22 28, 26,51 65-70 74

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