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Dr. Timo de Wolff Institute of Mathematics www.math.tamu.edu/~dewolff/Fall14/math302.html MATH 302 – Discrete Mathematics – Section 501 Final Exam Preparation Homework Fall 2014 Exercise 1. We denote all even integers by 2Z. Let for every j ∈ Z Ij denote the interval of real numbers from j/2 to j/2 + 1. For example, I5 = [2.5, 3.5]. Compute the following sets: S 1. j∈2Z (Ij ∩ Ij+1 ) S 2. j∈2Z (Ij \ Ij+1) S 3. j∈2Z (Ij ∩ Ij+1 ) ∩ (Ij \ Ij+1 ) S S 4. I ∩ I j j j∈2Z j∈Z\2Z Exercise 2. Solve the following counting problems: 1. In how many ways can five distinguishable hand holds be distributed to eight (distinguishable) spots on a climbing wall? 2. In how many ways can one order a green, a blue, and two black balls? 3. In how many ways can one form a (single) team of 4 persons out of 10 persons? 4. How many lists with 10 entries exist if every odd entry is a lowercase letter and every even entry a digit 0, 1, . . . , 9? 5. Recall the lizards in our garden (homework 8, exercise 3). Recently one of the lizards living under the house visited the two in the old tree and changed his color to brown, when he did. The other two lizards in the old tree decided since then not to take brown color. If the other assumptions from the exercise remain valid: how many combinations of colors can the 5 lizards have now? 6. Assume a group of 10 distinguishable persons vote for their favorite movie from a list of movies A, B, C, D. Assume the outcome is (4, 2, 1, 3). How many combinations of votes are possible, which turn into this result? 1 Exercise 3. Solve the following linear recurrence relation: an = −5an−1 − 6an−2 with initial conditions a0 = 2, a1 = 1. Exercise 4. Assume that containers with the amount of k ∈ N∗ TEU (twenty-foot equivalent units) were distributed to eleven ships at the Houston harbor on the last weekend. How large has k to be such that at least one ships carries containers of at least 5001 TEU? Prove your answer. Exercise 5. Prove by mathematical induction: 3 divides n3 + 2n for all n ∈ N∗ . Exercise 6. Show the following equation for all n ∈ N∗ : n X n j n 2 3 = j j=0 Exercise 7. Assume that three sets A, B, C satisfy #A = #B = #C = 10, #(A ∪ B ∪ C) = 21, #(A ∩ B) = 5 and A ∩ C = ∅. What is the cardinality of B ∩ C? Exercise 8. 1. Prove that the relation ∼ on R given by ∀a, b ∈ R : a ∼ b :⇔ ⌈a⌉ = ⌈b⌉ is an equivalence relation. 2. Prove that every equivalence class of ∼ has the same cardinality. 3. Let {2}∪{3}∪· · · ∪{12} be a partition of the set {2, . . . , 12}. Show that this partition is given by an equivalence relation, which can be defined by two dice. Compute the cardinalities of each equivalence set. 4. Let R be a relation on R2 (i.e., it is a subset of R2 × R2 ) such that for all pairs (a, b) and (c, d) holds: √ √ (a, b)R(c, d) :⇔ a2 + b2 ≥ c2 + d2 . Show that this relation is reflexive and transitive. Is it also antisymmetric (and hence a partial ordering)? How could the relation be changed slightly in order to make it asymmetric (this would prevent it from being reflexive anymore, of course)? Exercise 9. 1. Let A, B be two countable infinite sets and C, D two uncountable infinite sets. Assume you are allowed to construct a new sets out of A, B, C, D via using two intersections, one union and arbitrary many brackets. Will the resulting sets always be countable, uncountable or is both possible? Explain your answer. 2. Prove that Z2 (i.e., Z × Z) is countable. 2