[Discrete Mathematics] Direct Proofs Examples

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In this video we tackle a divisbility proof and then prove that all integers are the difference of two squares.

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Writing out each step more clearly (as most professors require) would improve your discrete tutorials. I appreciate your effort

TheMacTownPoke

Assume 7|4a where every a is an integer
7j = 4a. Because 7 is odd then j needs to be even (j = 2k).
7(2k) = 4a. Simplify
7k = 2a. Because 7 is odd then j needs to be even (k = 2r).
7(2r) = 2a. Simplify
7r = a
7|a

bous

How do you know to go down a path showing 5j=2a and then j=2k ? Why not one of the other paths I'm sure exists? Is it just practice that makes the path to the final proof more easy to find?

jamiewrithe

Prove using direct proof :
a.) If n is even and m is odd, then the product of nm is always even.
b.) Prove that, for any odd integer n, the number 2n2+5n+4 must be odd

sherlynrosales

For proof that 7 divides a, can we say that j = 4k since 4 is a multiple of 4, and therefore still true? Or could we not do this because not every even number can be written as 4k since numbers like 2 or 6 would require k to not be an integer?

Also, do you have videos on the symbols you use? I'm a bit unclear on what they mean and how to use them?

blakef.

Can you do it this way??
If 7|4a then there exists a number k (in the set of positive integers) such that 7k=4a;
If 7|a then there exists a number j (in the set of positive integers) such that 7j=a;

Then substituting a= 7j into 7k=4a,

7k=4(7j), then 7k=28j, which is k=4j.

If we substitute the value of k into 7k=4a, then 7(4j)=4a, which is 28j=4a

Thus 7j=a, proving 7|a.

andresnieves

what program do you use to make these tutorials??

BearfootBrad

Question: I enjoyed how you used a few examples to see a pattern and create a formula for such. My question is this: since I am not taking discrete mathematics currently (I am studying this independently), would I be allowed to do such thing in a formal proof? I feel like probably not, but how would you go about this if not? And how would I explain what I was doing? I am a physics major, and yet I'm required to still do proofs in physics, so I need to know for this semester.

bekkiiboo

Assume 7|4a.
Then 7j = 4a, and 4a/7 = j.
For 7|a, then it must be that a/7 = j/4.
For j to be an element of the positive integers, it must be that j is a multiple of 4.

bekkiiboo

Does anyone know what software he uses? It looks so clean I need it!

ebitdaddyca

Also a few examples of negative integers for the theorem "Every odd integer is a difference of two squares.":

2k + 1 = (k + 1)² - k²

1 = 1² - 0²
3 = 2² - 1²
5 = 3² - 2²
-1 = (0)² - (-1)² = 0² - 1²
-3 = (-1)² - (-2)² = 1² - 2²
-5 = (-2)² - (-3)² = 2² - 3²

✔

thepedzed

Prove 7|4a --> 7|a:
1. Assume 7|4a
2. 7j=4a=2(2a) [definition of divisibility: some j exists where 7j=4a; rewrite 4a as 2(2a) to demonstrate that it's even]
3. j=2k [j must be even to make 7j even because the 2 sides are equal, so j=2k for some k]
4. 7(2k)=2(2a) [substitute 2k for j]
5. 7k=2a [divide both sides by 2; now we can do steps 3-5 again essentially]
6. k=2m [LIKE STEP 3: k must be even to make 7k even, so declare m=2k for some m]
7. 7(2m)=2a [LIKE STEP 4: sub 2m for k]
8. 7m=a [LIKE STEP 5: divide both sides by 2]
9. 7|a [by definition of divisibility: some m exists where 7m=a, meaning 7|a]

grrrimamonster

In the Divisibility Examples video you showed a counter example to disprove a question similar to this. If a=2 wouldn't that be a counter example to 5|4?

brian_kirk