discrete math

show that the function f : x → y is surjective iff there exists a function h : y → x such that f · h = 1y

Step-by-step explanation:

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

Let us start by the first implication.

Our hypothesis is that the function is surjective. From this we know that for every there exist, at least, one such that .

Now, define the sets . Notice that the set is the pre-image of the element . Also, from the fact that is a function we deduce that , and because   the sets are no empty.

From each set  choose only one element , and notice that .

So, we can define the function as . It is no difficult to conclude that . With this we have that , and the prove is complete.

Now, let us prove the second implication.

We have that there exists a function    such that .

Take an element , then . Now, write and notice that . Also, with this we have that .

So, for every element we have found that an element (recall that ) such that , which is equivalent to the fact that is surjective. Therefore, the prove is complete.

step-by-step explanation:

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answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.