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In the first paragraph, we set up a proof that A ⊆ D ∪ E by picking an arbitrary x ∈ A. In the second, we used the fact that A ⊆ B ∪ C to conclude that x ∈ B ∪ C. Proving that one set is a subset of another introduces a new variable; using the fact that one set is a subset of the other lets us conclude new things about existing ...
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Physics is concerned with describing the interactions of energy, matter, space, and time, and it is especially interested in what fundamental mechanisms underlie every phenomenon. The concern for describing the basic phenomena in nature essentially defines the realm of physics.
Physics (Boundless) 3: Two-Dimensional Kinematics. 3.2: Vectors. Expand/collapse global location. 3.2: Vectors. Page ID. Boundless. Components of a Vector. Vectors are geometric representations of magnitude and direction and can be expressed as arrows in two or three dimensions. learning objectives.
- Sets
- A =
- 3∈A, 8∈A and 20∈A
- D =
- Intersection
- B ∩ C =
- Union
- Why Is Set Theory Important?
A set is a collection of things, which could be numbers. Set Theory is the study of sets and the properties they have. The things that make up a set are called the elementsof the set. You may not realize it, but you interact with lots of sets everyday. Any elements with something in common are part of a set. For example, fruits are a set. Bananas, ...
The number of elements of A is a property of the set A. A has three elements: the numbers 3, 8 and 20. We can also show that something is an element of A using the symbol ∈. The symbol ∈ is like saying “belongs to”. So, for set A:
We would say it as “3 belong to set A, 8 belongs to set A and 20 belongs to set A.” An important thing about sets is that the order of the elements does not matter. For example: A = {3, 8, 20} is the same set as A= {8, 3, 20} Here are examples of other sets, from Mrs. Brown’s class B= {all of the people with brown hair} = {Roberta, Jessica, Abdul, ...
There is also a symbol to show that a set is a subset of another set. It is ⊆. The symbol ⊆ is like saying “is a subset of". So, for our example above:
As you might guess, sometimes sets have some elements in common. The intersection of two sets is a set of all the elements that the two sets have in common. The symbol for intersection is ∩ . The symbol ∩is like saying “and”. Let’s look at an example. Let’s take the sets of people from Mrs. Brown’s class (remember sets B and C). The intersection of...
Here is another example that uses numbers: X = {0, 8, 2, 100} Y = {1, 12, 50, 8} Z = {8, 12, 1, 42, -1}
Another interesting concept in Set Theory is union. The union of two sets is what you get when you combine two sets together. It's like adding them. But, if the sets have an element in common, for example if they both contain a 4, then the union of those sets will only have ONE 4, not two. The symbol for union is ∪. The symbol ∪ is like saying “and...
Set theory is seen as the foundation of all mathematics. Sets are the building blocks of mathematics. The language of sets can be used to describe all mathematical concepts. Linear algebra, graph theory, analysis, and number theoryare all areas of math that are built around sets. Set theory also plays an important role in computer programming. Sets...
Physics can be intimidating, and people often want to know about the prerequisites to start learning physics. Listed below is a rough guide for the bare minimum of what you should know before taking particular physics classes.
In this chapter, we use set theory in yet another important field: functional analysis. More specifically, we prove two Han-Banach theorems: how to extend a functional linearly and continuously, and how to separate two convex sets from each other by a hyperplane. Both have a lot of applications in physics, detailed elsewhere.
