Need help i dont get it <br> So please help me

Daniel [21]

Answer:

Step-by-step explanation:

As the statement is ‘‘if and only if’’ we need to prove two implications

$f : X \rightarrow Y$ is surjective implies there exists a function $h : Y \rightarrow X$ such that $f\circ h = 1_Y$ .
If there exists a function $h : Y \rightarrow X$ such that $f\circ h = 1_Y$ , then $f : X \rightarrow Y$ is surjective

Let us start by the first implication.

Our hypothesis is that the function $f : X \rightarrow Y$ is surjective. From this we know that for every $y\in Y$ there exist, at least, one $x\in X$ such that $y=f(x)$ .

Now, define the sets $X_y = \{x\in X: y=f(x)\}$ . Notice that the set $X_y$ is the pre-image of the element $y$ . Also, from the fact that $f$ is a function we deduce that $X_{y_1}\cap X_{y_2}=\emptyset$ , and because $f$ the sets $X_y$ are no empty.

From each set $X_y$ choose only one element $x_y$ , and notice that $f(x_y)=y$ .

So, we can define the function $h:Y\rightarrow X$ as $h(y)=x_y$ . It is no difficult to conclude that $f\circ h(y) = f(x_y)=y$ . With this we have that $f\circ h=1_Y$ , and the prove is complete.

Now, let us prove the second implication.

We have that there exists a function $h:Y\rightarrow X$ such that $f\circ h=1_Y$ .

Take an element $y\in Y$ , then $f\circ h(y)=y$ . Now, write $x=h(y)$ and notice that $x\in X$ . Also, with this we have that $f(x)=y$ .

So, for every element $y\in Y$ we have found that an element $x\in X$ (recall that $x=h(y)$ ) such that $y=f(x)$ , which is equivalent to the fact that $f$ is surjective. Therefore, the prove is complete.

3 0

3 years ago

Need help asap pllzzz will give brainliest

Greeley [361]

Answer:

d. $350.00

Step-by-step explanation:

A constant is a number that does not have any variables. The additional cost would depend on the number of persons at attendance so the cost would be $10.00p which is not the constant here. Thus, $350.00 is the only constant in this situation.

4 0

3 years ago

Which ordered pairs cause this relation not to be a function? why? {(0, 0.5), (0.5, 1), (3, 1), (1, 6), (3, 2)}?

frosja888 [35]

Answer:

(3, 1), (3, 2)

Step-by-step explanation:

Pairs that have the same x-value are not allowed in a relation that is a function.

3 0

3 years ago

15x=75<br> help me plsssssssssssss

stiks02 [169]

5 75 divided by 15 hope this helps for next time

4 0

3 years ago

A group of band instructors answered a survey about hours of rehearsal per week and number of competitions won. The graph shows

IRISSAK [1]

Answer: B. 31

Step-by-step explanation:

This linear regression was constructed by relating the hours practiced per week and the number of competitions won.

Going by this graph, the number of competitions they can expect to win at 5 practices a week is 31.

This is derived by looking for the point where 5 competitions on the x-axis intersects with the line. This point is at 31 competitions on the y axis which would make it the answer.

5 0

3 years ago

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Need help i dont get it So please help me