Math Is Fun Forum

Again we have a rogue 6 in C.

Bob

A set and its complement divide the 'universe' into two non overlapping sets with no elements left out of being in one or the other. So when you unite them you get everything, ie the universe.

And when you look to find what's in both a set and its complement you find nothing hence the empty set.

Normally when I'm showing union and intersection in a picture I draw two overlapping circles (inside a rectangle for the universe). But for a set and its complement a better picture is a rectangle for the universe divided by a single border making the two sets that don't overlap and together make the whole of the universe rectangle.

Bob

Yes, that's got it!

Bob

Yes that's right. I've never heard it called the roster method though.  Always good to learn something new.

Bob

It is correct.

Bob

I only have the same search as you.  The keyword can throw up loads of hits. The trick is to remember a word you used that is unlikely to be in anyone else's posts.

I tried 3 people A,B and C and got pages and pages. I think it was finding every post that had any of those in it so then I tried "3 people A,B and C" hoping to find only that specific sentence but still didn't get to it.

Bob

hi Turbo

Welcome to the forum and thanks for the puzzles.

Two things I need to check:

y/y isn't that just 1 ?

You've copied these from somewhere.  Is there a copyright issue?

Bob

hi Zach,

Welcome back!

I do remember that question but I cannot find your old username in our lists, nor the thread.  Ho hum, not to worry.

Bob

Even and Odd are just words for something.  So, having thought more about this, they are defined as having the properties ( 2n, and 2n+1) rather than axioms.

A major part of mathematics is concerned with building models to 'describe' something useful.  Euclidean geometry, for example, enables us to work with triangles and circles and angles (and lots more) and leads to useful stuff like trigonometry.  It's an idealised view because you cannot have a line with zero thickness, nor can you measure an angle with absolute accuracy.  Nevertheless, it's very useful.

Number theory is built upon axioms such as x + y = y + x where x and y are real numbers.  We can use the axioms to build theorems and then the theorems plus axioms to build new theorems.  Eventually you end up with calculus and complex numbers.

The M-W definition looks ok to me.

If you can prove an axiom (presumably from some of the other axioms) then it isn't needed as an axiom, so you could develop your model with one less axiom.

For example. One axiom is "There exists a number (let's call it p for the moment) such that n + p = n.  This number is called the additive identity.  There is no axiom stating that p is unique because we can prove it.

Proof: suppose there are two identities p1 and p2.  Consider the sum p1 + p2.  It equals p1 because p2 is an identity, but it also equals p2 because p1 is an identity, thus p1 + p2 = p1 = p2 so the two identities are the same number.

Bob

Whose shortest distance from the given line segment is 3cm

Did you consider the direction of your infinite line?  I have received a hyperwave transmission from Alpha Centauri complaining about an infinite line that keeps intersecting their planet (every 24 hours) and, crucially, is interfering with their reception of House* on the interstellar carrier transmitter/receiver.  They believe the line originated in your area.

* They are big fans of Hugh Laurie.

It may only be imaginary to you, but, in interstellar carrier transmission/reception, imaginary lines play a significant role.  Indeed, without complex numbers it's doubtful such things would be possible

Bob

Hhmmm. Very primitive. My dial doesn't need batteries and will function for thousands of years. Only snag; the sun has to be shining.

Bob