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Decide whether each logical expression is a tautology, contradiction or neither. (p ∨ q) ∨ (q → p)     [Q1] (p → q) ↔ (p ∧ ¬q)  [Q2] (p → q) ∨ p   [Q3] (¬p ∨ q) ↔ (¬p ∧ q)   [Q4]

Below is an outline of the steps of the proof that  is irrational. Put the steps in the outline in the correct order. (a) Since d2 is even, d is even. (b) Squaring both sides of the equation =n/d leads to n2 = 2d2. (c) Since n is even, and n2 = 2d2, d2 is also even. (d) Then can be written as the ratio n/d of two integers n and d ≠ 0, such that n and d have no common factors. (e) Since n and d are both even,  n and d are both divisible by 2. (f) This a contradiction as n and d have a common factor. Therefore  must be irrational. (g) Since n2 is even, n is also even. (h) Suppose that  is rational. Proof: Step 1 [Step1]; Step 2 [Step2]; Step 3 [Step3]; Step 4 [Step4]; Step 5 [Step5]; Step 6 [Step6]; Step 7 [Step7]; Step 8 [Step8]  

A group of students is selected from a class. Every student in the class is either in the 3rd grade or the 4th grade, and no student is in both the 3rd and the 4th grades. How many students must be selected in order to guarantee that at least five 3rd graders or at least five 4th graders are selected?  

How many solutions are there to the equation  x1 + x2 + x3 + x4 + x5 + x6 = 37, where each xi is an integer that satisfies xi > 1 ?  

License plate numbers in a certain state consists of seven characters. The first character is a digit (0 through 9). The next four characters are capital letters (A through Z) and the last two characters are digits. Therefore, a license plate number in this state can be any string of the form:  Digit-Letter-Letter-Letter-Letter-Digit-Digit 1. How many different license plate numbers are possible?(a) 103  . 264(b) 10 . 9 . 8 . 264(c) 10 . 9 . 8 . 26 . 25 . 24 . 23(d) none of the above [Q1] 2. How many license plate numbers are possible if no digit appears more than once? (a) 103  . 264(b) 10 . 9 . 8 . 264(c) 10 . 9 . 8 . 26 . 25 . 24 . 23(d) none of the above [Q2] 3. How many license plate numbers are possible if no digit or letter appears more than once?(a) 103  . 264(b) 10 . 9 . 8 . 264(c) 10 . 9 . 8 . 26 . 25 . 24 . 23(d) none of the above [Q3]    

1. Which of the following graphs have a Hamiltonian path but not a Hamiltonian circuit? (a) K34(b) Q35(c) K5,6(d) K6,6(e) W35(f) none of them [Q1] 2. Which of the following graphs have both an Eulerian circuit and a Hamiltonian circuit? (a) K34(b) Q35(c) K2,3(d) K6,8(e) Q24(f) none of them [Q2]  

Consider the following claim and the statements Claim: For every real number x, if 0 ≤ x ≤ 3, then 15 – 8x + x2 > 0 (a) 0 ≤ x ≤ 3  (b) x < 0 or x > 3 (c) 15 – 8x + x2 > 0 (d) 15 – 8x + x2 ≤ 0 i. What would be the starting assumption in a proof by contrapositive of the statement above? [ContrapositiveAssumption] ii. What would be proven in a proof by contrapositive of the statement, given the assumption? [ContrapositiveConclusion]

Set Theory Questions.   Indicate which of the following statements are true. For any three sets, A, B, and C, if A ⊆ B, then A × C ⊆ B × C.    [Q1] R2 ⊆ R3.   [Q2] A × (B ∪ C) = (A × B) ∩ (A × C) [Q3] (A ∪ B) – (A ∩ B) = A – B. [Q4] A ∩ (B – A) = ∅. [Q5] Cardinality of the power set of the set {∅} is 2. [Q6]

Students are allowed to use 3, 8 1/2 x11  pages, handwritten, typed, or combination of both, front and back for the final exam. 

using full German sentences, write a short essay, between 200 and 300 words, in which you answer the following: what was your opinion of the games? Do you think they are a good way to teach kids to recycle and separate trash? Why or why not? What would you do differently? What worked or did not work?