1. Using the set of real numbers as the universe of discourse, describe why the following are valid:
a) ∃x ∀y (xy = x ²)
b) ∀x ∃y (x² + 6xy + 9y² = 0)
c) ∃y ∀x (x + y > xy)
d) ∀y ∃x (y − x = xy² + 1)
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please do A-C thank you 4. Let the universe of discourse, U, be the set of all people, and let M(x,y) be "x is the mother of y." Which of the following is a true statement? Translate it into English. a. (Ex)u((Vy)u(M(x, y))) b. (Vy)u((Ex)u(M(x, y))) C. Translate the following statement into logical notation using quantifiers and the proposition M(x, y) : "Everyone has a maternal grandmother."
5. Suppose P(m,n) means “m>n”, where the universe of discourse for m and n is the set of POSITIVE integers. Find the truth value of each statement and explain your answer. NOTE: This is NOT exactly the same as the practice test. (a) (2 points) VxP(x,5) (b) (2 points) Vx3yP(x,y) (c) (2 points) ExWyP(x,y)
1 YUI I0YEA). Exercise 2.65. Convert the following statements into statements using only logical sym- bols. Assume that the universe of discourse is the set of re Tse of discourse is the set of real numbers. (a) There exists a number x such that x2 + 1 is greater than zero. (b) There exists a natural number n such that m2 = 36. (c) For every real number x, x2 is greater than or equal to zero.
Suppose that we add a new quantifier called exists unique to first- (d) order logic, using the symbol 3! to represent it. It means that there is exactly one element of the universe that satisfies the subsequent formula. In this question, variables will range over the universe of numbers. [1 mark] (0) Is 3!x. x + x 2 valid? Why? Give an example of a valid formula that uses the quantifier, and an example of an unsatisfiable one. Both must...
Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) ∈ R if and only if a) x + y = 0 b) x= ±y. c) x-y is a rational number. d) = 2y. e) xy ≥ 0. f) xy = 0. g) x=l. h) r=1 or y = 1
2. In this problem, the domain of discourse is the set of positive integers: {1, 2, 3, ...}. Which statements are true? If an existential statement is true, give an example. If a universal statement is false, give a counterexample. (a) ∀x(x 2 − 1 > 0) (b) ∀x(x 2 − x > 0) (c) ∃x(x 3 = 8) (d) ∃x(x + 1 = 0)
For each of the following relations on the set of all real numbers, decide whether or not the relation is reflexive, symmetric, antisymmetric, and/or transitive. Give a brief explanation of why the given relation either has or does not have each of the properties. (x, y) elementof R if and only if: a. x + y = 0 b. x - y is a rational number (a rational number is a number that can be expressed in the form a/b...
homo 2nd order linear equations is necessarily the number -b/2a)]. 1. Find the general solution to the following homogeneous differential equations. (a) y" - 2y + y = 0 (b) 9y" + 6y + y = 0 (c) 4y" + 12y +9y = 0 (d) y' - 6y +9y = 0 2. Solve the the following initial value problems. (a) 9y" - 12y + 4y = 0 with y(0) = 2 and y(0) = -1 (b) y' + 4y +...
Construct expansions in a two-individual universe of discourse for the following sentences: Predicate Logic Symbolization 1. (x)(Fx ⋅ Gx) 3. (x)[Fx (Gx ∨ Hx)] 5. (x) (Fx Gx) 7. (x)(Fx Gx) 9. (x)[Fx (Gx Hx)] 11. (∃x)[(Fx ⋅ Gx) ∨ (Hx ⋅ Kx)] 13. (∃x) [(Fx Gx) ∨ (Fx Hx)]
Proof by contradiction that the product of any nonzero rational number and any irrational number is irrational (Must use the method of contradiction). Which of the following options shows an accurate start of the proof. Proof. Let X+0 and y be two real numbers such that their product xy=- is a rational number where c, d are integers with d 0. Proof. Let x0 and y be two real numbers such that their product xy is an irrational number (that...