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Book of Proof?
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7 reviews of this product found across Reddit:
Book of proof is a more gentle introduction to proofs then How to Prove it.
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No bullshit guide to linear algebra is a gentle introduction to linear algebra when compared to the popular Linear Algebra Done Right.
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An Illustrated Theory of Numbers is a fantastic introduction book to number theory in a similar style to the popular Visual Complex Analysis.
If any of you like or prefer a visual understanding of A22 or A31/A37 content, or love intuitive, puzzling, mathematical philosophy then yes, follow, save and subscribe to:
https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw
If proofs and set theory are a sore point, then voila:
https://www.people.vcu.edu/~rhammack/BookOfProof/
If you like to get physical with your set theoretic notation, and down and devilishly creative with your proofs on paper with paper in your grasp, then Amazon says pay up:
There's this, The Book of Proof https://smile.amazon.com/dp/0989472108/ref=cm_sw_r_cp_awdb_t1_jW04BbNE8ZA19
You could try Book of Proof by Richard Hammack. I've never read Velleman so I can't directly compare, but it's free for pdf (link to author's site above) and quite cheap in paperback (~$15). I found the explanations quite clear, the examples well worked and the exercises plentiful and helpful. Amazon reviewers seem to like it as well.
Set is a collection that can contain anything you wish. It's denoted by curly braces.
{dog, cat, paper} is a set whose elements are dog, cat, paper, so this set has 3 elements. We usually give sets names so that we don't have to refer to the roster notation like {dog, cat, paper} every time we mention them. We can let A = {dog, cat, paper}. We say A has size 3 because it has 3 elements in it. Symbolically, |A| = 3. Another more precise word for "size" is "cardinality".
A = {dog, cat, paper} = {paper, dog, cat}. That is the order of elements in the set doesn't matter. Also, A = {dog, cat, paper} = {dog, paper, dog, cat, cat, cat} because we can't count the same element in the set more than once.
Another more famous set is - Z - the set of all integers: {...-4, -3, -2, -1, 0, 1, 2, 3, 4...}. It contains infinite number of integers that extend in either direction forever.
Subset is any set that contains fewer or the same number of the same elements as in the parent set. Let A = {1, 2, 3, 4}. Then {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4} are the subsets of A. Technically, the empty set is also a subset of A, but that's not that important here. {0}, {5, 6} are not the subsets of A. By the way, if A = {1, 2} and B = {3, 4}, then A and B are the same size: |A| = |B| = 2, but A =/=B.
Relation is, formally, a set, but that probably will complicate things here. Instead, let's say that relation is a machine that takes the elements of one set(call that "domain"), does some stuff(as specified by you or whoever) to the elements of the domain and spits out the results into another(call that "codomain"). In other words, a relation matches the elements in the domain to the ones in the codomain according to a predefined rule. Instead of "matches" we often say "maps".
Consider A = {1, 2, 3} and B = {3, 100, 500, 6, 400, 10000000, 7, 9}. Define relation R to be multiplication by 3. Then A is the domain of R and B is its codomain because R maps 1 to 3, 2 to 6 and 3 to 9. Any subset of codomain all of whose elements are mapped to by the relation R is called the range of R. Let T = {3, 6, 9}. Then T is the range of R because T is the subset of B.
Functions are a special kind of relation. So, the relation R above is a function.
Now suppose K = {1, 2, 3} and L = {6, 8, 10}. Define relation r as r(1) = 6, r(1) = 10, r(3) = 8. Remember we can define a relation however we want? But this relation r is not a function because
A function can't map more than one element in the domain to the same element in the codomain. Here you can see that r is mapping 1 to 6 and 1 to 10. To see this more clearly consider the relation of addition by 4. Let f be a relation defined as f(1) = 5, f(4) = 8, f(4) = 9 where J = {1, 4} and G = {5, 7, 9, 8} are the domain and codomain of f respectively. So, this relation is saying that 4 + 4 = 8 and 4 + 4 = 9 which is nonsense, right? So, f is a relation, but not a function.
For the relation r to be considered a function, it must map every element in K to L, but r leaves the element 2 out in the cold.
Here I didn't even begin to scratch the surface of the topic of functions, so the next step for you would be to look up these books:
- A little intro that will hopefully make things much clearer:
Book of Proof by Richard Hammack
2. Hardcore mode or everything you ever wanted to know about functions:
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