Prefix Math Course# 301 Title Discrete Mathematics For It Units (3) 3 Hours Lecture: Fill & Download for Free

Thanks for the A2A!On my own, I’m trying to learn Linear Algebra and Multivariable Calculus, but I’m procrastinating, which is kind of stupid because I’m procrastinating at something I’m trying to teach myself? I’m the strict teacher and the lousy student simultaneously. I’ll tell the story about my school math class though:Well, at the beginning of this year, my class was supposed to be an Algebra 2 class, but she held a pre-test to see what everyone knew and I passed.Side note: I would have gotten a 100, but she used [math]f\circ g[/math] to represent function multiplication, not composition, and she took off points for say “shifted” instead of “translated.” Gah.Anyway, she allowed me to move on to the higher class, but since it was already 4–5 days into school, so the Pre-Calc teacher had already pre-tested everyone so I wasn’t able to test out of Pre-Calc.I pre-used a lot of pre-prefixes.

The prefix “poly” means many. A polynomial by definition is a function given by one term or a sum of terms with real number coefficients where the power of each variable is a non-negative integer. The lowest degree polynomial would be zero degrees given by a constant (think of this as a constant times a variable to the zero power.) You're confused because by definition a monomial is technically a polynomial and it only has one term and poly-nomial means “many” terms. It's just that by definition. We as mathematicians have agreed to this because it fits our nice clean definition. If it makes you feel any better you can think of a monomial as one term plus an infinite number of zero terms. Memorize the definition. The ability to identify functions as polynomials will be useful in higher level math courses.

Not necessarily, but yes with probability 1.I realize this sounds confusing, so let me explain.Does an infinite sequence of coin flips necessarily contain “Heads”?No, not necessarily. There exists an infinite sequence of coin flips with no Heads: Tails, Tails, Tails, Tails, …But we’re talking about random coin flips. Does a random sequence of coin flips necessarily contain Heads?You see, “random” and “necessarily” don’t go well together. If something is necessarily true, then of course it’s true for any random sequence. But if something is not necessarily true, then it’s also not necessarily true for a random sequence. Random sequences, by their nature, could yield anything in the sample space. At best, we could say “a random sequence fails to satisfy the condition with a low probability”.When the sample space is infinite, like here, “low probability” could actually be zero. There are events in infinite sample spaces which could happen, but the probability of them happening is precisely [math]0[/math]. Not “almost [math]0[/math]" or “tends to [math]0[/math]": exactly [math]0[/math].Similarly, an event could have probability [math]1[/math] even if it’s not the entire sample space. The technical term is “almost surely”. Something happens almost surely if it has probability [math]1[/math], and once again: this is not a sequence of events with a probability that tends to [math]1[/math] as [math]n\to\infty[/math]. It’s an event in a specific probability space and its probability is [math]1[/math].So, back to the question. It can be interpreted in different ways, but I think the intended meaning is this: we’re choosing numbers from the set [math]\{0,1,2,3,\ldots,n\}[/math] uniformly and independently, creating an infinite sequence of random numbers. What’s the probability that the sequence contains every possible finite combination as a sequence of consecutive outcomes?And the answer is that this happens almost surely, meaning with probability [math]1[/math]. It doesn’t necessarily happen: there are infinitely many infinite sequences of [math]\{1,2,3\}[/math] which never contain, say, [math]123[/math] consecutively in that order. Here’s an example of a prefix of such a sequence:[math]12211331213223212212132231322111132332\ldots[/math]Even though there are infinitely many such “bad” sequences, their total probability (or measure) in the space [math]\{1,2,3\}^\N[/math] is [math]0[/math]. They form a negligible set.Similarly, there are infinitely many infinite sequences of [math]0[/math]'s and [math]1[/math]'s which never contain [math]010[/math], and once again the total measure of those sequences is [math]0[/math]. Almost surely (with probability 100%), an infinite sequence of coin flips will, at some point, yield Heads-Tails-Heads.