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S has , x S, [x] = the equivalence class containing x = {s S: s x}. X y if 3 (x – y), [2] = {2, 5, -1,…} but [2] also = [5] or [-1], etc. Each element in [2] satisfies 3 (x – y): 2-2 = 3 which divides 3, 5-2 = 3 which divides 3, -1-2 = -3 which divides 3, etc. Ex: Define on R 2. By (x, y) (a, b) if x 2 y 2 = a 2 b 2. Check that this is an equivalence relation. Reflexivity: (x, y) R 2. Does (x, y) (x, y)? Since x 2 y 2 = x 2 y 2, this is true. Symmetry: (x, y), (a, b) R 2. Does x 2 y 2 = c 2 d 2? Proof: S...

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Nutmeg | mstaylor2016.wordpress.com Reviews

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S has , x S, [x] = the equivalence class containing x = {s S: s x}. X y if 3 (x – y), [2] = {2, 5, -1,…} but [2] also = [5] or [-1], etc. Each element in [2] satisfies 3 (x – y): 2-2 = 3 which divides 3, 5-2 = 3 which divides 3, -1-2 = -3 which divides 3, etc. Ex: Define on R 2. By (x, y) (a, b) if x 2 y 2 = a 2 b 2. Check that this is an equivalence relation. Reflexivity: (x, y) R 2. Does (x, y) (x, y)? Since x 2 y 2 = x 2 y 2, this is true. Symmetry: (x, y), (a, b) R 2. Does x 2 y 2 = c 2 d 2? Proof: S...

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Nutmeg | mstaylor2016.wordpress.com Reviews

https://mstaylor2016.wordpress.com

S has , x S, [x] = the equivalence class containing x = {s S: s x}. X y if 3 (x – y), [2] = {2, 5, -1,…} but [2] also = [5] or [-1], etc. Each element in [2] satisfies 3 (x – y): 2-2 = 3 which divides 3, 5-2 = 3 which divides 3, -1-2 = -3 which divides 3, etc. Ex: Define on R 2. By (x, y) (a, b) if x 2 y 2 = a 2 b 2. Check that this is an equivalence relation. Reflexivity: (x, y) R 2. Does (x, y) (x, y)? Since x 2 y 2 = x 2 y 2, this is true. Symmetry: (x, y), (a, b) R 2. Does x 2 y 2 = c 2 d 2? Proof: S...

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mstaylor2016.wordpress.com

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Review | Nutmeg

https://mstaylor2016.wordpress.com/2016/04/22/review

S has , x S, [x] = the equivalence class containing x = {s S: s x}. X y if 3 (x – y), [2] = {2, 5, -1,…} but [2] also = [5] or [-1], etc. Each element in [2] satisfies 3 (x – y): 2-2 = 3 which divides 3, 5-2 = 3 which divides 3, -1-2 = -3 which divides 3, etc. Ex: Define on R 2. By (x, y) (a, b) if x 2 y 2 = a 2 b 2. Check that this is an equivalence relation. Reflexivity: (x, y) R 2. Does (x, y) (x, y)? Since x 2 y 2 = x 2 y 2, this is true. Symmetry: (x, y), (a, b) R 2. Does x 2 y 2 = c 2 d 2? You are ...

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mstaylor2016 | Nutmeg

https://mstaylor2016.wordpress.com/author/mstaylor2016

S has , x S, [x] = the equivalence class containing x = {s S: s x}. X y if 3 (x – y), [2] = {2, 5, -1,…} but [2] also = [5] or [-1], etc. Each element in [2] satisfies 3 (x – y): 2-2 = 3 which divides 3, 5-2 = 3 which divides 3, -1-2 = -3 which divides 3, etc. Ex: Define on R 2. By (x, y) (a, b) if x 2 y 2 = a 2 b 2. Check that this is an equivalence relation. Reflexivity: (x, y) R 2. Does (x, y) (x, y)? Since x 2 y 2 = x 2 y 2, this is true. Symmetry: (x, y), (a, b) R 2. Does x 2 y 2 = c 2 d 2? Proof: S...

3

Relations | Nutmeg

https://mstaylor2016.wordpress.com/2016/03/25/relations

R from a set A to a set B is any subset R AxB. Recall: a function f: A B is a relation f AxB where a A,! Rather than saying (a, b) f, we usually write f(a) = b. Definition: a function f: A B is two-to-two. If the following is always true: whenever f(x) = f(y), it follows that x = y. Note: no two inputs go to the same output. F(x) = x 2, f(1) = f(-1), but 1 -1, so this is not injective. Definition: a function f: A B is called onto. If b B, a A, f(a) = b. If there is a bijection between them. F-g = C, g-h ...

4

January | 2016 | Nutmeg

https://mstaylor2016.wordpress.com/2016/01

January 29, 2016. A truth sets is a set of all missing information that makes an open sentence true. ex: {x R. If we assign a capital letter to a statement, we can manipulate it multiple ways. For example:. P: 5 2, Q: the door is open. P Q: 5 2 and the door is open. The symbol, , represents “and,” also called a “logical connective.”. P: 5 2. P = (5 2) = 5 2. P is true when P itself is false. An english statement example is (everyone loves Casey) = at least one person doesn’t love Casey. January 28, 2016.

5

Equivalence Classes and Partitions | Nutmeg

https://mstaylor2016.wordpress.com/2016/04/08/equivalence-classes-and-partitions

Equivalence Classes and Partitions. Of a set S is a collection of disjoint subsets ui S such that the union of all the subsets equal S. Lemma (another term for claim): Equivalence classes are either disjoint or equal. Theorem: The equivalence classes of an equivalence relation on S always partition S. Proof: Our previous lemma shows that these classes are disjoint. Since, by reflexivity, every element s [S], every element is in some equivalence class. This implies the union of all [x] in S = S.

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mathematicalypse.wordpress.com

Examples of Differentiable Manifolds | Casey Douglas

https://mathematicalypse.wordpress.com/2015/05/22/examples-of-differentiable-manifolds

Armageddon serious about maff. Examples of Differentiable Manifolds. May 22, 2015. June 2, 2015. Dedicates a fair amount of its zero-th chapter to examples of (real) differentiable manifolds. I wanted to add a few more examples, in addition to repeating some of those already mentioned. First, of course, are the Euclidean spaces,. Each of these can be covered by the single open set. Is the identity function. However, it should be noted that there are. And this can be generalized to the spaces. With a simi...

mathematicalypse.wordpress.com

Pre-Calc Review Problems | Casey Douglas

https://mathematicalypse.wordpress.com/2014/07/15/pre-calc-review-problems

Armageddon serious about maff. July 15, 2014. July 15, 2014. 1 Expand the product. Are variables representing unknown real numbers. 2 Here is an expression involving the variable. A) Is the expression. Equal to the original one? Why or why not? B) Is the expression. Equal to the original one? Why or why not? C) Is the expression. Equal to the original one? Why or why not? D) Is the expression. Equal to the original one? Why or why not? E) Are there any numbers that we are not allowed to plug in for.

mathematicalypse.wordpress.com

Quotient Manifolds | Casey Douglas

https://mathematicalypse.wordpress.com/2015/05/27/quotient-manifolds

Armageddon serious about maff. May 27, 2015. July 15, 2015. Quotient spaces are fundamental objects of study in mathematics, but they nonetheless feel strange or unclear the first few (dozen? Equivalence Relations and Quotient Spaces. Be a set equipped with an equivalence relation. Note that we needn’t impose any additional structure on. For what follows —. Is not assumed to be a group, a ring, a manifold, a vector space, or even a topological space, its just a set. The. The three defining properties.

mathematicalypse.wordpress.com

Tangent Vectors (Part I) | Casey Douglas

https://mathematicalypse.wordpress.com/2015/07/17/tangent-vectors-part-i

Armageddon serious about maff. Tangent Vectors (Part I). July 17, 2015. July 21, 2015. As discussed in our differential geometry course, a vector field,. On a differentiable manifold,. Is a correspondence that associates to each point. Is the velocity vector. This definition works perfectly fine when. Is an abstract manifold, though, we do not know how to differentiate the outputs of a curve. Even if we use the charts and local. Is a function with domain. This function is obtained from a curve. And we ha...

mathematicalypse.wordpress.com

Blog | Casey Douglas | Armageddon serious about maff

https://mathematicalypse.wordpress.com/mathematicalypse/blog

Armageddon serious about maff. October 27, 2016. The set of all points. Forms a sphere-like object called an ellipsoid centered at the point. A picture of this surface is shown below). Find the points on this ellipsoid that are closest to and farthest away from the point the origin. Find the equation for the tangent plane to the surface determined by. Find the maximum and minimum values of. Over all points in the region of space. Find the linear approximation of the function. At the point $latex (0, 1)4.

mathematicalypse.wordpress.com

More Great Math Quotes | Casey Douglas

https://mathematicalypse.wordpress.com/2013/01/28/more-great-math-quotes

Armageddon serious about maff. More Great Math Quotes. January 28, 2013. January 30, 2013. These are taken from various FOM students. 8220;Math was here before we were here to describe it.” – Cyrus. 8220;… the marks that dictate a mathematicians train of thought are far from meaningless … math is a necessary field of learning because it describes and proves why the physical world works the way it does.” – Kyle. 8221; – Cara. 8220;If x is an element of the null set, then this blog will be the best.&#8...

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mstaylor2016.wordpress.com

Nutmeg

S has , x S, [x] = the equivalence class containing x = {s S: s x}. X y if 3 (x – y), [2] = {2, 5, -1,…} but [2] also = [5] or [-1], etc. Each element in [2] satisfies 3 (x – y): 2-2 = 3 which divides 3, 5-2 = 3 which divides 3, -1-2 = -3 which divides 3, etc. Ex: Define on R 2. By (x, y) (a, b) if x 2 y 2 = a 2 b 2. Check that this is an equivalence relation. Reflexivity: (x, y) R 2. Does (x, y) (x, y)? Since x 2 y 2 = x 2 y 2, this is true. Symmetry: (x, y), (a, b) R 2. Does x 2 y 2 = c 2 d 2? Proof: S...

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