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  A 55-years-old woman who underwent hysterectomy for endometrial cancer. The structure measured is:

A student club holds a meeting. The predicate M(x) denotes whether person x came to the meeting on time. The predicate O(x) refers to whether person x is an officer of the club. The predicate D(x) indicates whether person x has paid his or her club dues. The domain is the set of all members of the club. The names of the members and their truth values for each of the predicates is given in the following table. Indicate whether each expression is true or false. Name M(x) O(x) D(x) Hillary T F T Bernie F T F Donald F T F Jeb F T T Carly F T F   ∀x ¬(O(x) ↔ D(x))   [Q1] ∀x ((x ≠ Jeb) → ¬(O(x) ↔ D(x)))   [Q2] ∀x (¬O(x) → D(x))   [Q3] ∃x (M(x) ∧ D(x))   [Q4] ∃x (O(x) → M(x))   [Q5] ∃x (M(x) ∧ O(x) ∧ D(x))   [Q6]  

How many solutions are there to the equation  x1 + x2 + x3 + x4 + x5 + x6 = 37, where each xi is a non-negative integer?  

Consider the problem of purchasing cans of juice. Suppose that juice is sold in 2-packs or 5-packs. Let Q(n) be the statement that it is possible to buy n cans of juice by combining 2-packs and 5-packs. This problem discusses using strong induction to prove that for any n ≥ 4, Q(n) is true. 1) The inductive step will prove that for k ≥ 5, Q(k+1) is true. What is the inductive hypothesis? (a) Q(j) is true for any j = 4, 5, …, k. (b) Q(j) is true for any j = 4, 5, …, k+1. (c) Q(j) is true for any j = 5, …, k+1. [Q1]   2) The inductive step will prove that for k ≥ 5, Q(k+1) is true. What part of the inductive hypothesis is used in the proof? (a) Q(k-2) (b) Q(k-1) (c) Q(k) [Q2]   3)In the base case, for which values of n should Q(n) be proven directly? (a) n = 3, n = 4, and n = 5. (b) n = 4 and n = 5. (c) n = 4, n = 5, and n = 6. [Q3]

Determine the truth value of each expression below. The domain is the set of all real numbers. ∀x∃y (xy > 0)  [Q1] ∃x∀y (xy = 0)  [Q2] ∀x∀y∃z (z = (x – y)/3)  [Q3] ∀x∃y∀z (z = (x – y)/3)  [Q4] ∀x∃y y2 = x   [Q5] ∀x∃y (x < 0 ∨ y2 = x)   [Q6] ∃x ∃y (x2 = y2 ∧ x ≠ y)  [Q7] ∀x ∀y (x2 ≠ y2 ∨ |x| = |y|)   [Q8]

Indicate whether each of the following arguments is valid or invalid.   If is an irrational number, then    is an irrational number. is an irrational number.∴   is an irrational number.   [Q1] p ↔ qp ∨ q∴ p   [Q2] ¬(p → q)q → p∴ ¬q    [Q3] q → pp∴ ¬(p → q)    [Q4] The patient has high blood pressure or diabetes or both.The patient has diabetes or high cholesterol or both.∴ The patient has high blood pressure or high cholesterol.  [Q5] ∀x (P(x) → Q(x))∃x ¬P(x)∴ ∃x ¬Q(x)   [Q6]

How many vertices does a full 4-ary tree with 50 internal vertices have?

The following two statements are logically equivalent.  If x is a rational number and y is an irrational number then x-y is an irrational number. If x is a rational number and x-y is an rational number then y is a rational number.

Below are the steps for a proof by contradiction of the following theorem: Theorem: There is no smallest positive real number. Put the steps of the proof in the correct order so that each step follows from previous steps in the proof. (a) This contradicts the assumption that r is the smallest positive real number. (b) Assume there is a smallest positive real number called r. (c) Consider r/2, which is a positive real number since r is positive. (d) Moreover, r/2 < r,  since both are positive. Proof: [Step1]; [Step2]; [Step3]; [Step4].    

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]