Define a function from N to Z that is both one to one and onto. Explain why it is a bijection?

Find a function from Q to Z that is one to one.

Please help me with these two questions. Thank you!

Find the set of ALL optimal solutions to the following LP: min z= 3x1−2x2 subject to 3x1+x2≤12 3x1−2x2−x3= 12 x1≥2 x1, x2, x3≥0

4. Translate each of the following statements into a symbolic logic and tell if each of the following is true or false, with a full justification (you do not have to justify your answer to (ii), which was done before) : (i) Every integer has an additive inverse. (ii) If a and b are any integers such that b > 0, then there exist integers q and r such that a = bq + r, where 0 ≤ r < b. (Note that this sentence does not have a uniqueness part of q or r.) (iii) Every integer has a unique multiplicative inverse. (Answer this question without using the symbol ∃! that we have not used much in class.) (iv) Any two real numbers x and y satisfy x < y. (v) Every real number has a greater real number. (vi) There exists a real number that is less than any real number. (vii) There are two real numbers x and y satisfy x < y. (viii) Given any two real numbers one of them is bigger than the other.

Write each vector as a linear combination of the vectors in S. (Use

s₁ and s₂,

respectively, for the vectors in the set. If not possible, enter IMPOSSIBLE.)

S = {(1, 2, −2), (2, −1, 1)}

(a)    z = (−13, −1, 1)

z =  

(b)    v = (−1, −5, 5)

v =  

(c)    w = (−2, −14, 14)

w =  

(d)    u = (1, −4, −4)

u =  

Analysis Integral confused concept. Please clear writing and follow the comment

1. upper sum with respect to P is U(f,P)= sum of Mk(xk-xk-1). Does that equal to U(f)=inf{u(f,P): p is a element of Q} (Q is the collection of all possible partition})

Also, is Mk=sup{f(x):x is a element of {xk-1,xk}?????

2. If Q is a refinement of P, then U(f, Q)<=U(f, P) and L(f, P),<=L(f,Q). I don't understand the geometric meaning behind this because Q has more element than P and contains all of the elements from P, However, why U(f, Q)<=U(f, P)????

This Q is not the Q from question1

1)  Recall, a truth table for a proposition involving propositional symbols p and q uses four rows for the cases p true, q true, p true, q false, p false, q true and p false, q false (in that order). For example  the outcome for p v ¬q  is  T, T, F, T  since the expression is only false when q is true but p is false. Of course, we have the same outcome for any logically equivalent proposition including ¬(¬p ∧ q), (¬p ∧ q) → false  and q → p. Of these, q → p clearly reduces the number of symbols to a bare minimum. Find "minimal expressions" for the other 15 possible outcomes, which are listed below:

a) FFFF

b) FFFT

c) FFTF

d) FFTT

e) FTFF

f) FTFT

g) FTTF

h) FTTT

i) TFFF

j) TFFT

k) TFTF

l) TFTT

m) TTFF

n) TTTF

o) TTTT

You may only use symbols from the set p, q, → , ∧ , ∨, (,  ),  ↔ , false,  true, ¬ .  Each of those count 1 toward the length of the expression. (Note:  falseand true still count as single symbols, even though they have multiple letters.) In some of the answers, your expression might have just p in it and not q or vice versa. Of course, there are also two answers that don't have either p or q in them!

(a) Find a 3×3 matrix A such that 0 is the only eigenvalue of A, and the space of eigenvectors of 0 has dimension 1. (Hint: upper triangular matrices are your friend!)

(b) Find the general solution to x' = Ax.

PLEASE SHOW YOUR WORK CLEARLY.

Show that the Legendre polynomials P1 and P2 are orthogonal by explicit integration. Also show that when (P2)^ 2 is integrated over the full range of integration, the result is 2 /(2l+1) , where l is the order of the polynomial.

Let f: A ->B and g:B -> A be functions. Prove that if fog is one-to-one and gof is onto, then f is a bijection.

Suppose there are two lakes located on a stream. Clean water flows into the first
lake, then the water from the first lake flows into the second lake, and then water from the second
lake flows further downstream. The in and out flow from each lake is 500 liters per hour. The first
lake contains 100 thousand liters of water and the second lake contains 200 thousand liters of water.
A truck with 500 kg of toxic substance crashes into the first lake. Assume that the water is being
continually mixed perfectly by the stream.

a) Find the concentration of toxic substance as a function of time in both lakes.
b) When will the concentration in the first lake be below 0.001 kg per liter?
c) When will the concentration in the second lake be maximal?

(a) Find all positive values of λ for which the following boundary value problem has a nonzero solution. What are the corresponding eigenfunctions? X′′ + 4Xʹ + (λ + 4) X = 0, X′(0) = 0 and X′(1) = 0. Hint: the roots of its auxiliary equation are –2 ± σi, where λ = σ².

(b) Is λ = 0 an eigenvalue of this boundary value problem? Why or why not?

How could I mathematically prove these statements?

1. The sum of the first n positive odd numbers is square.

2. Two positive numbers have the same set of common divisors as do the smallest of them and their absolute difference.

3. For every prime p > 3, 12|(p 2 − 1).

a) Let σ = (1 2 3 4 5 6) ∈ S6, find the cycle decomposition of σ i for i = 1, 2, . . . , 6.

(b) Let σ1, . . . , σm ∈ Sn be disjoint cycles. For 1 ≤ i ≤ m, let ki be the length of σi . Determine o(σ1σ2 · · · σm)

A circuit consisting of a resistor, capacitor and power supply is called an RC circuit. Physics and Kirchoff’s laws imply that if Q is the charge on the capacitor, R is the resistance and E is the power supply, then R(dQ/dt) + (1/C)Q = E. Let R = 20, C = .1 and E = 100e −.1t. If there is no charge on the capacitor at time t=0, find the charge Q at any time after that.

In an RL circuit, as described in class, R = .1, L = 1 and E = 10(1 − e −.05t ). If there is no current in the circuit at time t = 0, find the current at any time after that.

Solve the initial value problem dy/dt = 1 − 5y/(150 − 2t) , y(0) = 5. NOTE: Since we only care about what is going on up to when the tank is empty (i.e., t < 75 minutes), you can assume 150 − 2t > 0.