#12 in 7.5
It would be amazing if someone could explain the intersection between A and B on problem number 12. I have so far the squares being 31 and the cubes being 10. But I am not sure how to get the intersection. Any ideas? Also I do not understand 20 or 28 at all. If someone could please help me out that would ROCK! Thanks
Possible Solutions
I think that if you want numbers between 1 and 1,000 that are both cubes and squares you'd do (1,000)^1/6 but thats just my guess because when you want,
how many squares: (1,000)^1/2 = 31
how many cubics: (1,000)^1/3= 10
so how many squares and cubics: (1,000)^1/6 = 3
12. A: # of squares: 31 B: # of cubics: 10 A intersection B: 3
31+10-3=38
I also struggled with finding the intersection of the two, and although I got the same answer as you, I'm still lost. I need to know WHY you are choosing to raise 1000 to the power of $ \frac{1}{6} $.
Solution
We know that 1, 4, 9... are all squares, but we don't know where it stops. In other words, what number when squared, is the last square under 1000 (and thus we should stop counting)?
The answer to that question is $ a^2=1000 $. It's clear now that $ a=1000^\frac{1}{2} $=31.(some decimal) and so the largest square under 1000 is $ 31^2=961 $.
Similarly, the last cube under or equal to 1000 is 1000 itself, $ 10^3 $.
Also:
$ 1000^\frac{1}{5} $ = 3
&
$ 1000^\frac{1}{6} $ = 3
How do you know to take 1000 to the power of $ \frac{1}{6} $ and not $ \frac{1}{5}? $
Solution
By inclusion-exclusion, we need to take out the numbers we counted twice (ie numbers which are squares and cubes). Because 3 is not divisible by 2, you can get a numbers which is a square and cube by squaring some integer, and then cubing that result. For example, 2 squared is 4, and 4 cubed is 64. This can be written symbolically as $ (a^2)^3 $, where a is the integer being squared.
If you are like "ok, but how do I simplify that... struggling to remember exponent laws", just use the definition of exponent. No extra memorization required. $ a^2=a*a $. and $ b^3=b*b*b $. So $ (a^2)^3 = (a^2)*(a^2)*(a^2)=(a*a)*(a*a)*(a*a) $. Since integer multiplication is "associative", you can forget about parentheses. So it just becomes $ a*a*a*a*a*a $. There are 6 a there, so it equals $ a^6 $.
Using a similar approach to above, $ 1000^\frac{1}{6} $=3.something. (luckily, $ 1000^\frac{1}{5} $ also equals 3.something), so there are only 3 numbers counted twice.
#20 in 7.5
If anyone is still having trouble with 20, it might help to look at the answer to 19 in the back of the book. It gives an explicit formula for the number of elements in the union of 5 sets. It should help make the Principle of Inclusion-Exclusion a little more clear.
#28 in 7.5
Can someone rephrase the question or shed some light on what this question is asking? I looked at the solution for #29 which seems to be quite similar, but it was a notation we haven't learned in class.
I looked at the solution for #27, which seems to be a similar problem to #28. I think the inclusion/exclusion property for sets also holds for probabilities, and I think the point of the "no two events can occur at the same time" stipulation is to tell you that the probability of any intersection of events, whether 2, 3, or n, will be zero. That said, I think you just need to add the probabilities of the independent events.
Related: 4.1 Homework_MA375Fall2008walther
