Math Is Fun Forum

That is correct. As an illustration, let us apply this to the definition of continuity: let f(x) be a real-valued function of the real variable x. Then f is continuous iff the following holds:

where

denotes the statement

.

You should also be aware that

and

do not commute:

. In the first statement, the y depends on the particular x chosen; different choices of x may require different y. In the second statement, however, there is a unique y for all the x you choose.

For example, in the definition of continuity above, let us reverse the order of

and

:

This is no longer the definition of continuity, but of uniform continuity, which is a stronger condition than continuity.

Right again. And this time

commutes with

, as does

with

. For example, the definition of continuity above can be restated as follows:

It’s the same thing.

If B’s first statement is true, then A’s first statement has to be true as well, since the truth of “P and Q” implies that of “P or Q”. However one of them calls the other a liar; therefore A and B cannot both be knights. Therefore B’s first statement is false, i.e. B is a knave. Thus B’s second statement is also false. Hence, the capital is in the mountains but the road to the right does not go to the capital.

is by definition the additive inverse of

, i.e. the unique integer

such that

.

To prove that an integer is equal to

, add it to 15 and show that it equals zero.

Thus, to prove that

, you need to show that

. That is all the proof is doing. No other properties of

are assumed.

You can only factorize integers.

I get the following thumbnail but I can’t open it on the Google page.

Use Pythagoras’s theorem and this formula:

where d is semicircle diameter / triangle side and A and is semicircle area. Thus

Hence the area of the third semicircle is 20.

Let the centre of the circle be O.

Then we have OP = OQ (they’re radii of the circle) and ∠OPB =∠OQB = 90°. Hence BQ = BP.

Similarly CR = CP.

Therefore AB + BP = AQ = AR = AC + CP.

∴ AQ = AC + CP = ½(AC + CP + AC +CP) = ½(AB + BP + CP + AC) = ½ × perimeter.

To find the middle term of a G.P. with positive terms, you don’t need to find the common ratio: just multiply the two end terms and take the square root (i.e. take the geometric mean).

Similarly to find the middle term of an A.P., you don’t need the common difference: just add the two end terms and divide by 2 (arithmetic mean).

Faraday was probably one of these rare non-math scientists, in that case.

You need parentheses on the RHS. You probably mean

i.e. sum the RHS terms from left to right.

I remember reading somewhere that Michael Faraday was one of these big scientists who are mathematically illiterate – that all of his work on electricity was developed without using a single mathematical equation.