Suppose that x is real number. Prove that x+1/x =2 if and only
if x=1.
Prove that there does not exist a smallest positive real number.
Is the result still true if we replace ”real number” with
”integer”?
Suppose that x is a real number. Use either proof by
contrapositive or proof by contradiction to show that x3 + 5x = 0
implies that x = 0.
2. Prove the following properties.(b) Prove that x + ¯ xy = x + y.3. Consider the following Boolean function: F = x¯ y + xy¯ z +
xyz(a) Draw a circuit diagram to obtain the output F. (b) Use the
Boolean algebra theorems to simplify the output function F into the
minimum number of input literals.
Prove or disprove the statements: (a) If x is a real number such
that |x + 2| + |x| ≤ 1, then x 2 + 2x − 1 ≤ 2.
(b) If x is a real number such that |x + 2| + |x| ≤ 2, then x 2
+ 2x − 1 ≤ 2.
(c) If x is a real number such that |x + 2| + |x| ≤ 3, then x 2
+ 2x − 1 ≤ 2....
The payoff X of a lottery ticket in the Tri-State Pick 3 game is
$500 with probability 1/1000 and $0 the rest of the time. Assume
the payoffs X and Y are for separate days and are independent from
each other.
a. What price should Tri-State charge for a lottery ticket so
that they can break even in the long run (average profit =$ 0).
b. Find the mean and standard deviation of the total payoff
X+Y.
a) Prove that if X is Hausdorff, then X is T1
b) Give an example of a space that is T1 , but not
Hausdorff. Prove that the space you give is T1 and prove
it is not Hausdorff.
1. Prove that the Cantor set contains no intervals.
2. Prove: If x is an element of the Cantor set, then there is a
sequence Xn of elements from the Cantor set converging
to x.
Tri-X industries want to expand its production facilities at a
cost of $500,000. The equipment is expected to have an economic
life of 8 years, have a 7-year property class and have a resale
value of $55,000 after eight years of use. The annual operating
cost is expected to be $12,000 for the first year and to increase
by $750 per year thereafter. If the equipment is purchased, Tri-X
wants to compare the straight line depreciation method to the MACRS...
(a) Prove that there are no degenerate bound states in an
infinite (−∞ < x < ∞) one-dimensional space. That is, if
ψ1(x) and ψ2(x) are two bound-state solutions of − (h^2/2m) (d^2ψ
dx^2) + V (x)ψ = Eψ for the same energy E, it will necessarily
follow that ψ2 = Cψ1, where C is just a constant (linear
dependence). Bound-state solutions should of course vanish at x →
±∞.
(b) Imagine now that our particle is restricted to move...