Consider ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. i. Verify that d is really a metric. ii. State and prove the Bolzano-Weierstrass Theorem for ℝ3 with this metric.

Q: Let M = {0,1} and d be the discrete metric, then d is not complete %3D O True O False

A: Discrete metric space.

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Q: 16. The set S = { x∈R: x2 - 4<0} with the usual metric is

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Q: A metric space (X,d) such that the function d1(p, q) = (d(p, q)) i not a metric in X

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A: I have proved all four conditions of metric for d

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A: Let d metric on X. Here , d satisfies following properties ,  dx , y ≥ 0  ,  ∀x , y ∈ X dx , y = dy…

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Q: Let X = R, and d be the discrete metric, and A = [3, 6) Show that if A is compact or not

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Q: Is the set S = [0,1] with the discrete metric d separable? Explain

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Q: A of a metric space (x, d.)

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Q: 1. Let (S, d) be a discrete metric space and p E S. Identify Sı (p), S1 (p), S2(p). 2. Let R have…

A: We will use the definition of metric function to sketch.

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Q: If X is a metric space and Y C X, then Y is a metric space.

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Q: close ball is an open

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Q: 12. Show that d((x1, Y1), (x2, Y2)) = |x1x2- Y1Y2| is not a metric on X = R %3D %3D

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Q: An Elliptical norm for R2 is defined by + 4u2, where x = (1,u). (i) Prove that l is a norm for R2…

A: This is a problem of Functional Analysis.

Q: Given a set X with x, y ∈ X define ?0 ifx=y d(x,y)= 1 ifx̸=y Show that d is a metric

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Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.