Let P(x) = “x is blue.” Let Q(x) = “x is a kangaroo.” Let R(x) =...
Let P(x) = “x is blue.” Let Q(x) = “x is a kangaroo.” Let R(x) = “x can leap tall buildings in a single bound.” Let S(x) = “x wears a cape.”
Suppose that the domain consists of all animals.
a. Express each of the following statements using quantifies, logical connectives, and the functions defined above.
i. No kangaroos are blue.
ii. Some kangaroos wear capes.
iii. All animals that wear capes can leap tall buildings in a single bound.
iv. Blue kangaroos do not wear capes.
b. Determine if the following arguments are valid and explain why or why not.
i. If ii. and iii. from part (a) are true, then some kangaroos can leap tall buildings in a single bound.
ii. If i., iii. and iv. from part (a) are true, then blue kangaroos cannot leap tall buildings in a single bound.
Homework Answers
Let P(X) be the predicate " is a dragon." Let Q(x) be the predicate "x breathes...
Let P(X) be the predicate " is a dragon." Let Q(x) be the predicate "x breathes fire.” Let R(x,y) be the predicate "x and y are the same object.” Let S be an arbitrary nonempty set. Rewrite the following English statements in symbolic no- tation using predicates P, Q, R, universal and existential quanti- fiers, and any variables you want. i. There are no dragons in S. ii. Not everything in S is a dragon. iii. There is at least...
2(2) LetP(x); Q(x); R(x), andS(x) be the open sentences:P(x) :xis a baby.Q(x) :xis a logical.R(x) :xis...
2(2) LetP(x); Q(x); R(x), andS(x) be the open sentences:P(x) :xis a baby.Q(x) :xis a logical.R(x) :xis able to manage an alligator.S(x) :xis despised.Suppose that the universe,Uconsists of all people (and assume ther is some objective measure to determineeach of the sentences). Express each of these statements using quanti ers, logical connectives, and the aboveopen sentences. Then write the negation of the statement.(a) Babies are iillogical.(i) Statement in symbolic logic:(ii) Negation in symbolic logic:(iii) Negation in plain English
1) Let R be the relation defined on N N as follows: (m, n)R(p, q) if...
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Homework 19. Due April 5. Consider the polynomial p(z) = r3 + 21+1. Let F denote the field Q modu...
Homework 19. Due April 5. Consider the polynomial p(z) = r3 + 21+1. Let F denote the field Q modulo p(x) and Fs denote the field Zs[r] modulo p(x). (i) Prove that p(x) is irreducible over Q and also irreducible over Zs, so that in fact, F and Fs are fields (ii) Calculate 1+2r2-2r + in HF. (iii) Find the multiplicative inverse of 1 +2r2 in F. (iv) Repeat (ii) and (iii) for Fs. (v) How many elements are in...
2. Let e1(x) = 1, ez(x) = x, p1(x) = 1 – x and p2(x) =...
2. Let e1(x) = 1, ez(x) = x, p1(x) = 1 – x and p2(x) = –2 + x. Let E = (e1,e2) and B = (P1, P2). 2 a) Show that B is a basis for P1(R). 4 b) Let ce : R → R3 be the change of coordinates from E to ß. Find the matrix representation of C. Leave your answer as a single simplified matrix. 6 c) Let (:,:) be an inner product on P1(R). Suppose...
Let (X, d) be a metric space, f,g:X R some functions and xo e X,q E...
Let (X, d) be a metric space, f,g:X R some functions and xo e X,q E R. Assume that f(x) = g(x) whenever x € Bd (x.). PART I. Prove that if f(x) →q as x → Xo, then g(x) = q as x → Xo. PART II. Can we also conclude that if f(x) = q as x → 00, then g(x) →q as x → 00?
A. Let X be a binomial random variable with n = 74 and p = .6....
A. Let X be a binomial random variable with n = 74 and p = .6. Use the normal approximation to the binomial to find: (i) P(X ≤ 50) (iii) P(40 ≤ X ≤ 50) (v) P(X = 43) (ii) P(X ≥ 40) (iv) P(42 ≤ X < 49) B. Each time a roulette wheel is spun, there are 38 possible outcomes, 18 red, 18 black, and two green. Suppose that you ALWAYS bet "black". (i) Suppose the roulette wheel...
Question 1. Consider these real-valued functions of two variables TVIn (r2y2) f (x, y)- 9(r,)2 2+...
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(d) Let (x1,x) R..9x 2 yo} (3) S is the set of combinations of (x,x2) which produce at least output level yo.Econom...
(d) Let (x1,x) R..9x 2 yo} (3) S is the set of combinations of (x,x2) which produce at least output level yo.Economists refer to S'as the upper contour set associated with output yo. Assume that x (x,x2) S and y (y,y2) S. That is xfx yo and yy z yo. i) Let z (z1,z2) R.. What must be true for ze S? ( mark) ii) Let z= (z1,z2) x +(1A)y where 02<1 Prove that zE S Hint: Using results on...
Need help with this question. Thank you :) (6) (a) Consider the following graph P R...
Need help with this question. Thank you :)
(6) (a) Consider the following graph P R U T (i) What are the degrees of the vertices in the graph? (ii) Does the graph have a closed Euler trail? If so, give an example of a closed Euler trail in the graph. If not, explain why no closed Euler trail exists. (iii Give an example of a spanning tree in the graph (iv) Two identical looking bags are on a table....
Let P(X) be the predicate " is a dragon." Let Q(x) be the predicate "x breathes fire.” Let R(x,y) be the predicate "x and y are the same object.” Let S be an arbitrary nonempty set. Rewrite the following English statements in symbolic no- tation using predicates P, Q, R, universal and existential quanti- fiers, and any variables you want. i. There are no dragons in S. ii. Not everything in S is a dragon. iii. There is at least...
1) Let R be the relation defined on N N as follows: (m, n)R(p, q) if and only if m - pis divisible by 3 and n - q is divisible by 5. For example, (2, 19)R(8,4). 1. Identify two elements of N X N which are related under R to (6, 45). II. Is R reflexive? Justify your answer. III. Is R symmetric? Justify your answer. IV. Is R transitive? Justify your answer. V.Is R an equivalence relation? Justify...
Homework 19. Due April 5. Consider the polynomial p(z) = r3 + 21+1. Let F denote the field Q modulo p(x) and Fs denote the field Zs[r] modulo p(x). (i) Prove that p(x) is irreducible over Q and also irreducible over Zs, so that in fact, F and Fs are fields (ii) Calculate 1+2r2-2r + in HF. (iii) Find the multiplicative inverse of 1 +2r2 in F. (iv) Repeat (ii) and (iii) for Fs. (v) How many elements are in...
2. Let e1(x) = 1, ez(x) = x, p1(x) = 1 – x and p2(x) = –2 + x. Let E = (e1,e2) and B = (P1, P2). 2 a) Show that B is a basis for P1(R). 4 b) Let ce : R → R3 be the change of coordinates from E to ß. Find the matrix representation of C. Leave your answer as a single simplified matrix. 6 c) Let (:,:) be an inner product on P1(R). Suppose...
Let (X, d) be a metric space, f,g:X R some functions and xo e X,q E R. Assume that f(x) = g(x) whenever x € Bd (x.). PART I. Prove that if f(x) →q as x → Xo, then g(x) = q as x → Xo. PART II. Can we also conclude that if f(x) = q as x → 00, then g(x) →q as x → 00?
Question 1. Consider these real-valued functions of two variables TVIn (r2y2) f (x, y)- 9(r,)2 2+4 (a) (i) What is the maximal domain, D, for the functions f and g? Write D in set notation (ii) What is the range of f and g? Is either function onto? iii) Show that f is not one-to-one. (iv) Find and sketch the level sets of g with heights: z0 0, 20 2, 204 (Note: Use set notation, and draw a single contour...
(d) Let (x1,x) R..9x 2 yo} (3) S is the set of combinations of (x,x2) which produce at least output level yo.Economists refer to S'as the upper contour set associated with output yo. Assume that x (x,x2) S and y (y,y2) S. That is xfx yo and yy z yo. i) Let z (z1,z2) R.. What must be true for ze S? ( mark) ii) Let z= (z1,z2) x +(1A)y where 02<1 Prove that zE S Hint: Using results on...
Need help with this question. Thank you :)
(6) (a) Consider the following graph P R U T (i) What are the degrees of the vertices in the graph? (ii) Does the graph have a closed Euler trail? If so, give an example of a closed Euler trail in the graph. If not, explain why no closed Euler trail exists. (iii Give an example of a spanning tree in the graph (iv) Two identical looking bags are on a table....
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