Question

In: Advanced Math

1. When proving If p then q.” DIRECT PROOF you need to: CONTRAPOSITION you need to:...

1. When proving If p then q.”

DIRECT PROOF you need to:

CONTRAPOSITION you need to:

CONTRADICTION you need to:

2. Prove by direct proof that if m and n are integers, with m odd and n is even, then 5n + m2 is odd.

3. Prove by contraposition that if x 6= 5 and is irrational, then 4x x − 5 is irrational.

4. Prove the following existential statements by providing a value for x. In both cases, the universe is the set of all real numbers.

a) ∃x x 2 + 5x − 7 = 0

b) ∃x x < 10 → (x − 2) 2 < 0

5. Prove that for any integer n, there exists an even integer k so that n < k + 1 < n + 3.

6. Prove or disprove: If x is rational and y is irrational, then xy is irrational.

7. Prove that there is no positive integer n so that 49 < n 2 < 64.

8. Prove or disprove: ∀x∃y ((x − 3)y = 4x), where the universe of discourse is R for both variables.

9. Prove, by contraposition, that if the product of two real numbers is irrational, then at least one of the two numbers is irrational. (In other words: If x · y is irrational, then x is irrational OR y is irrational.”)

10. Prove, by contradiction, that √ 3 is irrational. You may use the Little Theorem: If m2 is a multiple of 3, then m itself is a multiple of 3

Expert Solution

Colby Messinger answered 2 years ago

give a constructive proof of fn = Q^n + P^n/ Q - P , where Q...

give a constructive proof of fn = Q^n + P^n/ Q - P , where Q is the positive root and P is negative root of x^2 - x - 1= 0 fn is nth term of fibonacci sequence, f1 = 1 f2, f3 = f2 +f1, ... fn= fn_1 +fn_2 , n>2

Give a proof or counterexample, whichever is appropriate. 1. NOT (∃x, (P(x) OR Q(x) OR R(x)))...

Give a proof or counterexample, whichever is appropriate. 1. NOT (∃x, (P(x) OR Q(x) OR R(x))) is logically equivalent to ∀x, ((NOT P(x)) AND (NOT Q(x)) AND (NOT R(x))). 2. NOT (∃x, (P(x) AND Q(x) AND R(x))) is logically equivalent to ∀x, ((NOT P(x)) OR (NOT Q(x)) OR (NOT R(x))). 3. NOT (∃x, (P(x) ⇒ Q(x))) is logically equivalent to ∀x, (P(x)⇒ NOT Q(x)). 4. NOT (∃x, (P(x) ⇒ Q(x))) is logically equivalent to ∀x, (P(x) AND (NOT Q(x))). 5....

1. Show that the argument (a) p → q q → p therefore p...

1. Show that the argument (a) p → q q → p therefore p V q is invalid using the truth table. ( 6 marks ) (b) p → q P therefore p is invalid using the truth table. ( 6 marks ) (c) p → q q → r therefore p → r is invalid using the truth table. ( 8 marks )

Profit-maximizing Q (quantity) and P (price) will you get a different Q and P if you...

Profit-maximizing Q (quantity) and P (price) will you get a different Q and P if you use equations 2 and 4 vs. equations 2, 3, and 5? (1) Demand: Q = 230 – 2.5P + 4*Ps + .5*I, where Ps = 2.5, I = 20. (2) Inverse demand function [P=f(Q)], holding other factors (Ps = 2.5 and I =20) constant, is, P=100-.4*Q. (3) Production: Q = 1.2*L - .004L2 + 4*K - .002K2; (4) Long Run Total Cost: LRTC =...

1. Give a direct proof that if n is an odd integers, then n3 is also...

1. Give a direct proof that if n is an odd integers, then n3 is also an odd integer. 2. Give a proof by contradiction that the square of any positive single digit decimal integer cannot have more than two decimal digits.

For a situation of a certain concave mirror creating an image of an object, when p=6cm, q=12cm. What is the value of p when q=-12cm?

For a situation of a certain concave mirror creating an image of an object, when p=6cm, q=12cm. What is the value of p when q=-12cm?Group of answer choices-6cm.-3cm.3cm.6cm.15cm.

1. Suppose that the market demand is described by P = A – B(Q+q) where P...

1. Suppose that the market demand is described by P = A – B(Q+q) where P is the market price, Q is the output of the incumbent firm, and q is the output of the potential entrant to the market. The incumbent’s total cost function is C(Q) = c1Q, whereas the cost function of the entrant is C(q) = c2q+F. a. If the entrant firm observes the incumbent producing Q* units of output and expects this output level to be...

1.) Suppose that the statement form ((p ∧ ∼ q)∨(p ∧ ∼ r))∧(∼ p ∨ ∼...

1.) Suppose that the statement form ((p ∧ ∼ q)∨(p ∧ ∼ r))∧(∼ p ∨ ∼ s) is true. What can you conclude about the truth values of the variables p, q, r and s? Explain your reasoning 2.Use the Laws of Logical Equivalence (provided in class and in the textbook page 35 of edition 4 and page 49 of edition 5) to show that: ((∼ (p ∨ ∼ q) ∨ (∼ p ∧ ∼ r)) ∧ s) ≡ ((r...

For p, q ∈ S^1, the unit circle in the plane, let d_a(p, q) = min{|angle(p)...

For p, q ∈ S^1, the unit circle in the plane, let d_a(p, q) = min{|angle(p) − angle(q)| , 2π − |angle(p) − angle(q)|} where angle(z) ∈ [0, 2π) refers to the angle that z makes with the positive x-axis. Use your geometric talent to prove that d_a is a metric on S^1.

1. Determine if the following deduction rule is valid: p∨q ¬p _______ ∴ q 2. Determine...

1. Determine if the following deduction rule is valid: p∨q ¬p _______ ∴ q 2. Determine if the following is a valid deduction rule: (p∧q)→r ¬ p ∨ ¬ q ________ ∴ ¬r 3. Suppose p and q are (possibly molecular) propositional statements. Prove that p and q are logically equivalent if any only if p↔q is a tautology.