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3.1: The idea of a relation
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Cool_Brisk_Walk_Through_Discrete_Mathematics_(Davies)/03%3A_Relations/3.1%3A_The_idea_of_a_relation
I listed them out methodically to make sure I didn’t miss any (all the Harry’s first, with each drink in order, then all the Ron’s, etc.) but of course there’s no order to the members of a set, so I c...I listed them out methodically to make sure I didn’t miss any (all the Harry’s first, with each drink in order, then all the Ron’s, etc.) but of course there’s no order to the members of a set, so I could have listed them in any order. Think it out: every one of the ordered pairs in $X \times Y$ either is, or is not, in a particular relation between $X$ and $Y$ .
4.3: Philosophical interlude
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Cool_Brisk_Walk_Through_Discrete_Mathematics_(Davies)/04%3A_Relations/4.3%3A_Philosophical_interlude
But if we flip it a million times and get 500,372 heads, we can confidently say that the probability of getting a head on a single flip is approximately .500. For the Bayesian, the probability of some...But if we flip it a million times and get 500,372 heads, we can confidently say that the probability of getting a head on a single flip is approximately .500. For the Bayesian, the probability of some hypothesis being true is between 0 and 1, and when an agent (a human, or a bot) makes decisions, he/she/it does so on the most up-to-date information he/she/it has, always revising beliefs in various hypotheses when confirming or refuting evidence is encountered.
6.4: Summary
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Cool_Brisk_Walk_Through_Discrete_Mathematics_(Davies)/06%3A_Structures/6.4%3A_Summary
$n^{\underline{k}}$ — this is when we’re taking a sequence of $k$ different things from a set of $n$ , but no repeats are allowed. (A special case of this is $n!$ , when $k=n$ .) If not (and mo... $n^{\underline{k}}$ — this is when we’re taking a sequence of $k$ different things from a set of $n$ , but no repeats are allowed. (A special case of this is $n!$ , when $k=n$ .) If not (and most trainers would probably recommend against such monomaniacal approaches to excercise) then the real answer is only $18^{\underline{6}}$ , since we have 18 choices for our first block, and then only 17 for the second, 16 for the third, etc. (This reduces the total to 13,366,080.)
5: Structures
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Cool_Brisk_Walk_Through_Discrete_Mathematics_(Davies)/05%3A_Probability
Back Matter
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Cool_Brisk_Walk_Through_Discrete_Mathematics_(Davies)/zz%3A_Back_Matter
3.1.2: Binary (base 2)
https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Professor_Holz'_Topics_in_Contemporary_Mathematics/03%3A_Number_Bases_and_Modular_Arithmetic/3.01%3A_Binary_Numbers/3.1.02%3A_Binary_(base_2)
Start with the binary representation of $90_{10}$ : \[\begin{array}{*{8}{c@{\hspace{0pt}}}} \texttt{0} & \texttt{1} & \texttt{0} & \texttt{1} & \texttt{1} & \texttt{0} & \texttt{1} & \texttt{0} \\ \e...Start with the binary representation of $90_{10}$ : $\begin{array}{*{8}{c@{\hspace{0pt}}}} \texttt{0} & \texttt{1} & \texttt{0} & \texttt{1} & \texttt{1} & \texttt{0} & \texttt{1} & \texttt{0} \\ \end{array}$ Flip all the bits to get: $\begin{array}{*{8}{c@{\hspace{0pt}}}} \texttt{1} & \texttt{0} & \texttt{1} & \texttt{0} & \texttt{0} & \texttt{1} & \texttt{0} & \texttt{1} \\ \end{array}$ and finally add one to the result: \[\begin{array}{*{9}{c@{\hspace{0pt}}}} & & & & & & & \texttt{1} & …
2.10: Subsets
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Cool_Brisk_Walk_Through_Discrete_Mathematics_(Davies)/02%3A_Sets/2.10%3A_Subsets
Sometimes the expression “ $a \in X$ " is pronounced “ $a$ is an element of $X$ ," but other times it is read “ $a$ , which is an element of $X$ ". This may seem like a subtle point, and I guess it...Sometimes the expression “ $a \in X$ " is pronounced “ $a$ is an element of $X$ ," but other times it is read “ $a$ , which is an element of $X$ ". This may seem like a subtle point, and I guess it is, but if you’re not ready for it it can be a extra stumbling block to understanding the math (which is the last thing we need).
5.2: Trees
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Cool_Brisk_Walk_Through_Discrete_Mathematics_(Davies)/05%3A_Probability/5.2%3A_Trees
Suppose we chose A as the root of Figure $\PageIndex{1}$ . Then we would have the rooted tree in the left half of Figure $\PageIndex{2}$ . The A vertex has been positioned at the top, and everythi...Suppose we chose A as the root of Figure $\PageIndex{1}$ . Then we would have the rooted tree in the left half of Figure $\PageIndex{2}$ . The A vertex has been positioned at the top, and everything else is flowing under it. Every tree has a root except the empty tree, which is the “tree" that has no nodes at all in it. (It’s kind of weird thinking of “nothing" as a tree, but it’s kind of like the empty set $\varnothing$ , which is still a set.)
2.11: Power sets
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Cool_Brisk_Walk_Through_Discrete_Mathematics_(Davies)/02%3A_Sets/2.11%3A_Power_sets
Example: suppose $A$ = { Dad, Lizzy }. Then the power set of $A$ , which is written as “ $\mathbb{P}(A)$ " is: { { Dad, Lizzy }, { Dad }, { Lizzy }, $\varnothing$ }. Take a good look at all those...Example: suppose $A$ = { Dad, Lizzy }. Then the power set of $A$ , which is written as “ $\mathbb{P}(A)$ " is: { { Dad, Lizzy }, { Dad }, { Lizzy }, $\varnothing$ }. Take a good look at all those curly braces, and don’t lose any. As a limiting case (and a brain-bender) notice that if $X$ is the empty set, then $\mathbb{P}(X)$ has one (not zero) members, because there is in fact one subset of the empty set: namely, the empty set itself.
3.3: Relations between a set and itself
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Cool_Brisk_Walk_Through_Discrete_Mathematics_(Davies)/03%3A_Relations/3.3%3A_Relations_between_a_set_and_itself
Note that just because a relation’s two sets are the same, that doesn’t necessarily imply that the two elements are the same for any of its ordered pairs. Surely all three wizards have looked in a mir...Note that just because a relation’s two sets are the same, that doesn’t necessarily imply that the two elements are the same for any of its ordered pairs. Surely all three wizards have looked in a mirror at some point in their lives, so in addition to ordered pairs like (Ron, Harry) the hasSeen relation also contains ordered pairs like (Ron, Ron) and (Hermione, Hermione).
Index
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Cool_Brisk_Walk_Through_Discrete_Mathematics_(Davies)/zz%3A_Back_Matter/10%3A_Index

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