(1)Prove that for every a, b ∈ R, |a + b| = |a| + |b| ⇐⇒...

(1)Prove that for every a, b ∈ R, |a + b| = |a| + |b| ⇐⇒ ab ≥ 0. Hint: Write |a + b| 2 = (|a| + |b|) 2 and expand.

(2) Prove that for every x, y, z ∈ R, |x − z| = |x − y| + |y − z| ⇐⇒ (x ≤ y ≤ z or z ≤ y ≤ x). Hint: Use part (1) to prove part (2).

Expert Solution

Colby Messinger answered 2 years ago

a. Prove that y=sin(x) is a subspace of R^2 b. Prove that a set of 2x2...

a. Prove that y=sin(x) is a subspace of R^2 b. Prove that a set of 2x2 non invertible matrices a subspace of all 2x2 matrices

1. Prove or disprove: if f : R → R is injective and g : R...

1. Prove or disprove: if f : R → R is injective and g : R → R is surjective then f ◦ g : R → R is bijective. 2. Suppose n and k are two positive integers. Pick a uniformly random lattice path from (0, 0) to (n, k). What is the probability that the first step is ‘up’?

Let x, y ∈ R. Prove the following: (a) 0 < 1 (b) For all n...

Let x, y ∈ R. Prove the following: (a) 0 < 1 (b) For all n ∈ N, if 0 < x < y, then x^n < y^n. (c) |x · y| = |x| · |y|

b）Prove that every metric space is a topological space. (c) Is the converse of part (b)...

b）Prove that every metric space is a topological space. (c) Is the converse of part (b) true? That is, is every topological space a metric space? Justify your answer

Prove 1. For each u ∈ R n there is a v ∈ R n such...

Prove 1. For each u ∈ R n there is a v ∈ R n such that u + v= 0 2. For all u, v ∈ R n and a ∈ R, a(u + v) = au + av 3. For all u ∈ R n and a, b ∈ R, (a + b)u = au + bu 4. For all u ∈ R n , 1u=u

Question 1. Equivalence Relation 1 Define a relation R on by iff . Prove that R...

Question 1. Equivalence Relation 1 Define a relation R on by iff . Prove that R is an equivalence relation, that is, prove that it is reflexive, symmetric, and transitive. Determine the equivalence classes of this relation. What members are in the class [2]? How many members do the equivalence classes have? Do they all have the same number of members? How many equivalence classes are there? Question 2. Equivalence Relation 2 Consider the relation from last week defined as:...

Let R=R+. Define: a+b = ab ; ab = a^(lnb) 1. Is (R+, +, ) a...

Let R=R+. Define: a+b = ab ; a*b = a^(lnb) 1. Is (R+, +, *) a ring? 2. If so is it commutative? 3. Does it have an identity?

prove by using induction. Prove by using induction. If r is a real number with r...

prove by using induction. Prove by using induction. If r is a real number with r not equal to 1, then for all n that are integers with n greater than or equal to one, r + r^2 + ....+ r^n = r(1-r^n)/(1-r)

1. Let A be an inductive subset of R. Prove that {1} ∪ {x + 1...

1. Let A be an inductive subset of R. Prove that {1} ∪ {x + 1 | x ∈ A} is inductive. 2. (a) Let n ∈ N(Natural number) and suppose that k 2 < n < (k + 1)2 for some k ∈ N. Prove that n does not have a square root in N. (b) Let c ∈ R \ {0}. Prove that if c has a square root in Z, then c has a square root in...

5. (a) Prove that the set of all real numbers R is uncountable. (b) What is...

5. (a) Prove that the set of all real numbers R is uncountable. (b) What is the length of the Cantor set? Verify your answer.

Question