Consider the lattice of real numbers in the interval [0,1] with the relation ≤. Does this...

Consider the lattice of real numbers in the interval [0,1] with the relation ≤. Does this lattice have any atoms?

Expert Solution

Colby Messinger answered 1 year ago

verify the assertion. (Subspace example) 1) The set of continuous real-valued functions on the interval [0,1]...

verify the assertion. (Subspace example) 1) The set of continuous real-valued functions on the interval [0,1] is a subspace of R^[0,1] This is from Linear Algebra Done Right- Sheldon Axler 3rd edition. I don't understand why the solution uses a integral.

The Cantor set, C, is the set of real numbers r for which Tn(r) ϵ [0,1]...

The Cantor set, C, is the set of real numbers r for which Tn(r) ϵ [0,1] for all n, where T is the tent transformation. If we set C0= [0,1], then we can recursively define a sequence of sets Ci, each of which is a union of 2i intervals of length 3-i as follows: Ci+1 is obtained from Ci by removing the (open) middle third from each interval in Ci. We then can define the Cantor set by C= i=0...

Suppose we define a relation ~ on the set of nonzero real numbers R* = R\{0}...

Suppose we define a relation ~ on the set of nonzero real numbers R* = R\{0} by for all a , b E R*, a ~ b if and only if ab>0. Prove that ~ is an equivalence relation. Find the equivalence class [8]. How many distinct equivalence classes are there?

Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric,...

Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x,y) ∈ R if and only if: a) x = 1 OR y = 1 b) x = 1 I was curious about how those two compare. I have the solutions for part a) already.

Real numbers p and q are randomly chosen from the interval 0 to 1, inclusive. If...

Real numbers p and q are randomly chosen from the interval 0 to 1, inclusive. If r is given by r = 2(p + q), and p, q, r are rounded to the nearest integers to give P, Q and R, respectively, determine the probability that R = 2(P + Q). (As an example, if p=0.5 and q=0.381, then r=1.762, and so P =1, Q=0, and R=2)

Consider the equivalence relation on Z defined by the prescription that all positive numbers are equivalent,...

Consider the equivalence relation on Z defined by the prescription that all positive numbers are equivalent, all negative numbers are equivalent, and 0 is only equivalent to itself. Let f ∶ Z → {a, b} be the function that maps all negative numbers to a and all non-negative numbers to b. Does there exist a function F ∶ X/∼→ {a, b} such that f = F ○ π? If so, describe it .

3. In an FCC lattice, the dispersion relation of phonons may be taken to be identical...

3. In an FCC lattice, the dispersion relation of phonons may be taken to be identical to the 1-D model, with equilibrium separation a being equal to the distance between close packed nearest neighbours. Calculate the force constant of gold. Given atomic weight = 197 g mol−1, and density = 1.93 × 10^4 kg m^−3. The speed of sound in gold = 3.24 × 10^3 m s^−1.

2. Define a relation R on pairs of real numbers as follows: (a, b)R(c, d) iff...

2. Define a relation R on pairs of real numbers as follows: (a, b)R(c, d) iff either a < c or both a = c and b ≤ d. Is R a partial order? Why or why not? If R is a partial order, draw a diagram of some of its elements. 3. Define a relation R on integers as follows: mRn iff m + n is even. Is R a partial order? Why or why not? If R is...

On the set S of all real numbers, define a relation R = {(a, b):a ≤ b}. Show that R is transitive.

3) Prove that the cardinality of the open unit interval, (0,1), is equal to the cardinality...

3) Prove that the cardinality of the open unit interval, (0,1), is equal to the cardinality of the open unit cube: {(x,y,z) E R^3|0<x<1, 0<y<1, 0<Z<1}. [Hint: Model your argument on Cantor's proof for the interval and the open square. Consider the decimal expansion of the fraction 12/999. It may prove handdy]

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