Match the verbal expression (term) with its algebraic expression (definition)

Mathematics

1 answer:

Liula [17]3 years ago

5 0

A variable cubed D) x³

Quotient of some number and three E) b ÷ 3

Product of an unknown value and three B) 3a

Three less than a variable A) y - 3

Three more than some number C) z + 3

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A space is totally disconnected if its connected spaces are one-point-sets.Show that a finite Hausdorff space is totally disconn

marysya [2.9K]

Step-by-step explanation:

If X is a finite Hausdorff space then every two points of X can be separated by open neighborhoods. Say the points of X are $x_1, x_2, ..., x_n$ . So there are disjoint open neighborhoods $U_{12}$ and $U_2$ , of $x_1$ and $x_2$ respectively (that's the definition of Hausdorff space). There are also open disjoint neighborhoods $U_{13}$ and $U_3$ of $x_1$ and $x_3$ respectively, and disjoint open neighborhoods $U_{14}$ and $U_4$ of $x_1$ and $x_4$ , and so on, all the way to disjoint open neighborhoods $U_{1n}$ , and $U_n$ of $x_1$ and $x_n$ respectively. So $U=U_2 \cup U_3 \cup ... \cup U_n$ has every element of $X$ in it, except for $x_1$ . Since $U$ is union of open sets, it is open, and so $U^c$ , which is the singleton $\{ x_1\}$ , is closed. Therefore every singleton is closed.

Now, remember finite union of closed sets is closed, so $\{ x_2\} \cup \{ x_3\} \cup ... \cup \{ x_n\}$ is closed, and so its complemented, which is $\{ x_1\}$ is open. Therefore every singleton is also open.

That means any two points of $X$ belong to different connected components (since we can express X as the union of the open sets $\{ x_1\} \cup \{ x_2,...,x_n\}$ , so that $x_1$ is in a different connected component than $x_2,...,x_n$ , and same could be done with any $x_i$ ), and so each point is in its own connected component. And so the space is totally disconnected.

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4 years ago

Ashley is the utilities manager at a paper mill. She's watching over one of her holding tanks. It takes of an hour to fill the t

Romashka [77]

If it takes an hour to fill 1 tank in 1 hour 1 tank will be full

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4 years ago

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Let f(x) = √x and g(x) = x be two functions over the set of negative real numbers. Find (f+g)x ,(f-g)x , (fg)x , (f/g)x .

Delicious77 [7]

$RELATIONS \: \: AND \: \: FUNCTIONS \\ \\ \\ We \: have \: , \: \\ \\ \\ (f+g)(x ) \: = \: \sqrt{x} + x \\ \\ \\ (f - g)(x ) \: = \: \sqrt{x} - x \\ \\ \\ (fg)(x) = \sqrt{x} (x) = {x}^{ \frac{3}{2} } \\ \\ \\ (\frac{f}{g} )(x) = \frac{ \sqrt{x} }{x} = {x}^{ \frac{ - 1}{ \: \: \: 2} } \: \: \: \: \: \: \: \: \: Ans.$