Well-defined sets and the Barber Paradox

For a set to be well-defined, an object must be in the set or not in the set.  For example, the set of rugby players at StFX is well-defined.  The set of good rugby players at StFX is not well-defined because the notion of "good" is up for debate, it is subjective.  We want to avoid situations like that.

There is a famous paradox about a barber that also illustrates what can happen if a set is not well-defined.

Youtube video about Barber paradox

Website explaining the Barber Paradox

Units Digt in Powers of 7 question

I didn't do the best job of explaining this example in class so I thought I would try to explain a similar example that will hopefully make things a bit clearer.

Question: Find the units digit of 7 raised to the 362.

Solution:

First- figure out what the question is asking.  7 raised to the 362 means 7 times itself 362 times, so you can't do that on a calculator.  The "units digit" is just asking what digit is in the ones position.

Devise a plan.  Look for patterns in powers of 7.

Carry out the plan: 

7^0 = 1, 7^1 = 7, 7^2 = 49, 7^3 = 343
7^4 = 2401, 7^5 = 16,807, 7^6 = 117,649, 7^7 = 823,543

We notice that the units digit follows a pattern: 1, 7, 9, 3, 1, 7, 9, 3,...
The patterns repeats itself in blocks of 4.

362 divided by 4 is 90.5.  360 is the closest multiple of 4 to 362 that is also smaller than 362.

So the units digit of 7^360 is 1 (because it is the first in the block of four)

The units digit of 7^361 is 7 (because it is the second in the block of 4)

The units digit of 7^362 is 9 (because it is the third in the block of 4) [and this is the final solution]

Note: there is no way to check our solution because a calculator won't give a number this big.

First Post!

Okay, this is going to be a pretty feeble attempt to get the ball rolling.  Tomorrow I will mention that we use deductive reasoning when we apply Pythagoras' theorem to solve for a length in a specific right triangle.  For example, if we know that a right triangle has sides with length 12 and 5, then the hypotenuse must have length = 13 because 12^2 + 5^2 = 13^2.  The real reason I want to mention this is because of something cool I learned earlier this year.  The Flatiron Building in New York City is a right triangle, and its sides are in the proportions 12-5-13.  The Museum of Mathematics in NYC held an event on Dec 5, 2013 (12-5-13) to demonstrate this.  They even used glow-sticks!  Here are two links about it:

Pythagorazing the Flatiron

Glow Sticks Prove Math Theorem

I went to New York this summer and of course had to go to the Museum of Math MoMath, it was awesome.  It is just off Broadway, and I could see the Flatiron Building as I walked to the museum from the subway station.  Considering how large New York is, that is pretty cool.   One of my research areas is fractal trees, and the museum had a fun activity that turned me into a fractal tree.