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
In: Advanced Math
A 1000L tub containing red and white plastic beads is initially full and 30% of the beads are red. Some workers are constantly dumping buckets of beads (2L size, containing half red and half white) into the tub at a rate of 1 bucket every 30 seconds. Other workers are continuously stirring the beads and, as necessary, removing excess volume.
(a) Write an IVP which describes this situation.
(b) Solve the IVP.
(c) How much (units are liters) of red and white beads will be in the tank after an hour?
(d) In the long term, how many white beads (in volume) will be in the tank?
In: Advanced Math
Let G be an abelian group and K is a subset of G.
if K is a subgroup of G , show that G is finitely generated if and only if both K and G/K are finitely generated.
In: Advanced Math
Q1. Arrival of customers to a local store may be modeled by a Poisson process with an average of 1 arrival every 10 min period. Each customer on average stays for a time exponentially distributed with mean 15 minutes. This is modeled as a birth-death process.
(i) What assumption is necessary for this process to be modeled as a birth-death process?
(ii) Compute the probability that the number of customers in the store reaches 30, if we have 3 customers at the start.
Pls explain with workings. Thxs
In: Advanced Math
GIVEN: COS(x) +3xe^-x=0 USING NEWTON RAPHSON METHOD Find: 1.) The POSITIVE ROOT USING X0=2 2.) THE NEGATIVE ROOT USING X0=-0.5 *STOPPING CRITERION ≤ 0.01% use radian mode in calcu and i dont want a program answers pls i need the manual method.
In: Advanced Math
Let x = inf E. Prove that x is an element of the boundary of E. Here R is regarded as a metric space with d(p,q) = |q−p| for p,q ∈R
In: Advanced Math
Show your understanding of the unsolved mathematical below and present a description of the problem.
Yang-Mills Theory and the Mass Gap (mathematical physics)
In: Advanced Math
For each of the following sequences find a functionansuch that the sequence is a1, a2, a3, . . ..
You're looking for a closed form - in particular, your answer may NOT be a recurrence (it may not involveany otherai). Also, while in general it is acceptable to use a "by cases"/piecewise definition, for this task you must instead present a SINGLE function that works for all cases.(Hint: you may find it helpful to first look at the sequence of differences of consecutive terms.)
1) 7,1/11, 15,1/19, 23,1/27, 31,. . .
2) 1, 2, 2, 3, 3, 3, 4, 4, 4, 4,. . .where each i∈N is repeated i times. (Hint: one or more of the following may be useful: floor function, Gauss sum, quadratic formula)
In: Advanced Math
Find A, B, C, D, E such that the rule Af(a − 2h) + Bf(a − h) + Cf(a) + Df(a + h) + Ef(a + 2h) approximates f"'(a) with the error O(h^p) for the largest order of convergence p. What is this p equal to? Please be clear and show all steps, thanks!
In: Advanced Math
Find A, B, C, D, E such that the rule Af(a − 2h) + Bf(a − h) + Cf(a) + Df(a + h) + Ef(a + 2h) approximates f"'(a) with the error O(h^p) for the largest order of convergence p. What is this p equal to? Please be clear and show all steps, thanks!
In: Advanced Math
As a property owner, you want to fence a garden which is adjacent to a road. The fencing next to the road must be stronger and cost $6 per foot. The fencing on the other sides cost $4 per foot. The area of garden is 2400 square feet.
1. Draw several diagrams to express the situation and calculate the cost for each configuration, and then estimate the dimension of minimum cost.
2. Find the function that represents the cost in terms of one of its sides.
3. Using the graphing utility, graph this cost function
4. Find the dimension that minimizes the cost of fencing and compare this with your estimate based the diagram you made.
In: Advanced Math
The stem direction of a note starts to point downward when the note is on the _____ line of the staff? (Hint: the bottommost line is the first line)
1. |
second |
|
2. |
third |
|
3. |
fourth |
|
4. |
fifth |
In: Advanced Math
FOR THE PARAMETRIZED PATH r(t)= e^tcos(πt)i+e^tsin(πt)j+e^tk
a) find the velocity vector, the unit tangent vector and the arc lenght between t=0 and t=1
b) find a point where the path given by r(t) intersects the plane x-y=0 and determine the angle of intersection between the tangent vector to the curve and the normal vector to the plane.
In: Advanced Math
Consider an initial value problem
?′′ + 2? = ?(?) = cos? (0 ≤ ? < ?) , 0 (? ≥ ?)
?(0) = 0 and ?′(0) = 0
(a) Express ?(?) in terms of the unit step function.
(b) Find the Laplace transform of ?(?).
(c) Find ?(?) by using the Laplace transform method.
In: Advanced Math
1. Prove that (a) ⊆ C(a). Conclude that (a) ≤ C(a)
2. Prove that for each a ∈ G, Z(G) ⊆ C(a). Conclude that Z(G) ≤ C(a).
In: Advanced Math