Clicker Question. Theorems/Proofs and Computational Problems/Algorithms MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES

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1 MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES Tuesday, 1/21/14 General course Information Sets Reading: [J] 1.1 Optional: [H] Exercises: Do before next class; not to hand in [J] pp : 1, 3, 5, 31, 32, 41, 45, 57, 59, 84, 87, 88 Always feel free to ask about exercises at the next class! Some exercises might be assigned for presentation or hand-in. Tuesday, 1/21/14, Slide #1 Clicker Question I am (or plan to be) a major/minor in: A. Mathematics only (not CS) B. Computer Science only (not math) C. Both Math and CS D. Neither Math nor CS Tuesday, 1/21/14, Slide #2 Theorems/Proofs and Computational Problems/Algorithms In both math and computer science we ask questions and try to find answers, but with different points of view. Math: Given a conjecture, we try to prove or disprove it. If it s proved true, it s a theorem. Example: Theorem. There are infinitely many prime numbers. CS: Given some input and a task, we try to design an algorithm that we can program to perform the task correctly for any particular input. Example: Input: A list of numbers. Task: Create a new list containing these numbers, sorted into increasing order. Tuesday, 1/21/14, Slide #3 1

2 Common Aspects of Math and CS Math: The proof of a theorem may actually be an algorithm. CS: To be sure an algorithm always works, we must be able to prove that it does. Basic objects and methods: Both math and CS use the language of set theory to describe the objects used. [J] 1.1, [H] Chap. 1 Both math and CS use rules of logic to reason correctly in designing proofs and algorithms. [J] , [H] Chap. 2 Tuesday, 1/21/14, Slide #4 Sets: Fundamental Objects of Math and CS A set is any collection of objects: We can list the elements between curly brackets: {North, South, East, West} = {East, South, West, North} List order doesn t matter! We can use set builder notation: {n n is an odd number and 1 n 10} ={1,3,5,7,9} Especially useful for describing sets that are hard to list, e.g. infinite sets The smallest set has no elements: Notation: «= { } = empty set Sets can have elements that are themselves sets: {{Black, White}, {High, Low}, {Loud, Quiet}} {«} (This is not the empty set!} Tuesday, 1/21/14, Slide #5 Some Important Sets Z = {, -2, -1, 0, 1, 2, } The Integers. N = {1, 2, } The Natural Numbers Q = The Rational Numbers R = The Real Numbers C = The Complex Numbers Tuesday, 1/21/14, Slide #6 2

3 Clicker Exercise Let S = {n n is in N, and { < }. Written as a list, the set S is: A. { 4, 3, 2, 1,0,1,2,3,4} B. {0,1,2,3} C. {1,2,3} D. {0,1,2,3,4} E. {1,2,3,4} Tuesday, 1/21/14, Slide #7 Notation: Membership, Containment, Equality, Cardinality (Size) x A A B A = B x is an element of set A A is a subset of B A equals B A B A is a proper subset of B A B and A B A The cardinality of A For a finite set A, A = # of elements in A N = Tuesday, 1/21/14, Slide #8 Union and Intersection of Sets A B The union of A and B, the set containing all elements that are in A or in B, or in both. A B A B = { x ( x A) or ( x B) } The intersection of A and B, the set containing all elements that are in both A and B A B = { x ( x A) and ( x B) } QUESTION: How are A B and A B related to A and B? Tuesday, 1/21/14, Slide #9 3

4 Venn Diagrams: Diagrams to help visualize set intersections Universe: Large rectangle indicating all possible elements Sets: Regions inside universe Elements: Points inside regions QUESTION: Below, A is circle on left and B is circle on the right. What color regions are: A B? A B? Complement of A, denoted or A c, = A c = {x x A} Difference A B, A B = {x x ΠA and x B} Tuesday, 1/21/14, Slide #10 Clicker Question In the Venn Diagram shown, the element 5 is in A. Sets A, B, and C U = {n 1 <= n <= 10} A B C B. Sets A and B only C. Sets A and C only D. Only Set A E. Only Set C Tuesday, 1/21/14, Slide #11 Venn Diagrams: Examples With =C, the complex numbers, draw a Venn Diagram showing the relationship between the sets Z (the integers), N (the natural numbers), Q (the rational numbers), and R (the real numbers. With U = {1,2,..., 15}, draw a Venn Diagram showing the relationship between: E = even #s < 10 O = odd #s < 10 P = prime #s < 10 T = all #s less than 5 Tuesday, 1/21/14, Slide #12 4

5 General Venn Diagrams for 3 Sets The diagram below has a region for each possible intersection of 1 or more of three sets A, B, C. This can be used to illustrate some basic set identities. Examples: U The associative laws: = = De Morgan s Law s: = = A C B Tuesday, 1/21/14, Slide #13 Set Identities (pp. 8-9 in [J]) Tuesday, 1/21/14, Slide #14 Power Sets Let A be a set. The power set of A, denoted (A) or 2 A, is the set of all subsets of A. What are the power sets of: A = {0} A = {x, y} A = {1, 2, 3} A = «For a finite set A, how is (A) related to A? Tuesday, 1/21/14, Slide #15 5

6 Cartesian Products If S and T are sets, then the Cartesian Product of S and T is the set S x T, defined by: S T = {( s, t) s S and t T} Let A = {a, b, c}, B = {a, x, y, b} What is A x B? Is it the same as B x A? What is A 2 = A x A? For finite sets S and T, how is S x T related to S and T? Tuesday, 1/21/14, Slide #16 Clicker Question Let S = {u, v, w} and T = {7, 11}. Which of the following is a subset of S x T? A. The set S B. (u, 7) C. {{v, 7}, {w, 11}} D. {(u, 11), (v, 11), (w, 11)} E. {(v, 7), (11, w)} Tuesday, 1/21/14, Slide #17 6