Final Prep Flashcards
Explain how Parmenides proved dragons exist
Parmenides could declare something to be adragon To debate it
you would need to say that’s not what A DRAGON IS. Hence you
declare dragons exist
Explain how Zeno would contend that there is only one one in the entire universe.
He would say, ‘If I humor you and there are at least two points, we can draw a line between the two. A line is made of points, each of which has no length. If you remove one, you’ve removed no length. If you remove more, you’ve still removed no length. When you’ve removed them all, you’ve repeatedly removed nothing. So there’s no length left, and nothing was removed, so there was never a line to begin with.
Explain how Zeno would contedn that a fast Achilles could not catch a slow turtle in a head start
Every time Achilles catches up to where the turtle was, the turtle moves ahead. No matter how many times Achilles does this, the turtle is always ahead; he can never catch it.
Assume you have 40 Vertices in a graph and you want to check ever potential Hamiliton circuit. How many different circuits would you need to check. If you check 1000 per second and there been about 4.7 x 10^17 seconds in the life of the universe so far, why is there a problem?
We may assume we start at 1. A circle starts anywhere. Then there are 39 choices for the 1st, 38 for the 2nd. So, there are 39! choices and 38! for the second.
What are the advantages of using formal math over intuitive math?
Formal Math is more precise
What are the advantages of using intuitive math over formal math?
intuitive math is easier to explain
How can “formal” math be misleading?
Just cuz it’s symbols doesn’t mean it’s formal math
What happens to the notation and proofs of a mathematical subject as it ages? Explain
it gets more precise as time goes on
Addition Rule Assumptions
The sets don’t overlap
The sets include all elements
Multiplication/Power Rule Assumptions
Every combination is allowed.
Each selection is for something distinct
Why is it impossible for you to create a list of lists that don’t contain themselves?
If such a set exists, it leads to a contradiction when we ask whether it contains itself or not.
What does this evaluate to:
5
Σ (i^(2) + 1)
i=3
5
Σ (i^(2) + 1) = (3^2 + 1) + (4^2 + 1) + (5^2 + 1)
i=3
what does this evaluate to:
Σ (10i + 1)
i∈{1, 3, 10}
Σ (10i + 1) = (10(1) + 1) + (10(3) + 1) + (10(10) + 1)
i∈{1, 3, 10}
what does this evaluate to:
ℿ (n+1)
i∈{1, 3, 6}
ℿ (n+1) = (1+1) * (3+1) * (6+1)
i∈{1, 3, 6}
if B_1 = {1,3,6} and B_3 = {4, 10} and B_2 = {1,3}, what does this evaluate to:
⋃ B_i
B_2
B_1 ∪ B_3 = {1,3,6} ∪ {4, 10} = {1,3,4,6,10}
what is Σ (f(a) + g(a)) equal to?
Σ f(a) + Σ g(a)
what is Σ cf(a) equal to?
c Σ f(a)
what is ΣΣ f(x) equal to?
Σ(Σ f(x))
How do you prove a function is one to one?
Show f(x) = f(y)
What is Saturation (Sat)
If two edges are already used at a
vertex, the remaining edges can’t be used since a circuit can only visit a vertex once.
What is Over Saturation (OSat)?
If three edges at a vertex must be used, then the graph can’t have a Hamilton circuit. Clearly this is because the circuit can only visit
a vertex once. Over saturation is used at the end of a proof or case since it causes a contradiction to the existence of the circuit
what is Undersaturation (USat)?
This is similar to degree 2 except it causes a contradiction since there are not enough edges at the vertex to form the circuit.
Small Circuit (SC)
If some of the must use edges form a circuit that doesn’t hit every vertex, then the graph can’t have a Hamilton circuit. This is simply
because a circuit does not contain any smaller circuits as subgraphs.
Why must isomorphism be one to one and onto?
It makes sure that the graph is the same always. We can reverse it will still have the exact same connection between the vertexs and will look the same.
what is permutative symmetry in a few sentences and in laymans terms?
think of it as a quality where you can shuffle or rearrange components of an object, and it still looks the same
