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A set is a collection of things, usually numbers. We can list each element (or "member") of a set inside curly brackets like this: Common Symbols Used in Set Theory. Symbols save time and space when writing. Here are the most common set symbols. In the examples C = {1, 2, 3, 4} and D = {3, 4, 5}
- Mathematical Symbols
Symbols save time and space when writing. Here are the most...
- Real Numbers
Any point on the line is a Real Number: The numbers could be...
- Power Set
You will see in a minute why the number of members is a...
- Algebraic Number
So the imaginary number i is an Algebraic Number. Note:...
- Introduction to Sets
What is a set? Well, simply put, it's a collection. First we...
- Symbols in Algebra
set symbols (curly brackets) {1,2,3} = equals: 1+1 = 2: ≈ :...
- Sets and Venn Diagrams
Sets. A set is a collection of things. For example, the...
- Set-Builder Notation
Type of Number. It is also normal to show what type of...
- Mathematical Symbols
- Sets
- Operations
- Binary Operations
- Well Defined
- Introduction to Groups
- Formal Definition of A Group
- Only Two Operations
- Why Groups?
- Special Types of Groups: Abelian
Before reading this page, please read Introduction to Sets, so you are familiar with things like this: 1. Set of clothes: {hat, shirt, jacket, pants, ...} 2. Set of even numbers: {..., -4, -2, 0, 2, 4, ...} 3. Positive multiples of 3 that are less than 10: {3, 6, 9}
Now that we have elements of sets it is nice to do things with them. Specifically, we wish to combine themin some way. This is what an operation is used for. An operation takes elements of a set, combines them in some way, and produces another element. or, more simply: An operationcombines members of a set.
So far we have been a little bit too general. So we will now be a little bit more specific. A binary operation is just like an operation, except that it takes 2 elements, no more, no less, and combines them into one. You already know a few binary operators, even though you may not know that you know them: 1. 5 + 3 = 8 2. 4 × 3 = 12 3. 4 − 4 = 0 The...
One thing about operators is that they must be well defined. But reverse that. They must be defined well. Think about applying those two words, "defined well" to the English language. If a word is defined well, you know exactly what I mean when I say it. 1. The word "angry" is defined pretty well, as you know exactly what I mean when I say it. 2. B...
Now that we understand sets and operators, you know the basic building blocks that make up groups. Simply put: A group is a set combined with an operation So for example, the set of integers with addition. But it is a bit more complicated than that. We can't say much if we just know there is a set and an operator. What more could we describe? We ne...
Let's look at those one at a time: 1. The group contains an identity.If we use the operation on any element and the identity, we will get that element back. For the integers and addition, the identity is "0". Because 5+0 = 5 and 0+5 = 5 In other words it leaves other elements unchanged when combined with them. There is only one identity element for...
Way back near the top, I showed you the four different operators that we use with the numbers we are used to: But in reality, there are only two operations! When we subtract numbers, we say "a minus b" because it's short. But what we really mean is "a plus the additive inverseof b". The minus sign really just means add the additive inverse. But it ...
So why do we care about these groups? Well, that's a hard question to answer. Not because there isn't a good one, but because the applications of groups are very advanced. For example, they are used on your credit cards to make sure the numbers scanned are correct. They are used by space probes so that if data is misread, it can be corrected. They ...
If a * e = a, doesn't that mean that e * a = a? And similarly, if a * b = e, doesn't that mean that b * a = e? Well, as a matter of fact, it does. But we are careful here because in general, it is not true that a * b = b * a. But when it is true that a * b = b * a for all a and b in the group, then we call that group an abeliangroup. That fact is t...
The number of elements in a finite set \(A\) is called its cardinality, and is denoted by \(|A|\). Hence, \(|A|\) is always nonnegative. If \(A\) is an infinite set, some authors would write \(|A|=\infty\); however, we will use more specific designations for the cardinality of infinite sets.
Set notation is mathematical notation that is used in set theory and probability. A set can be a list of items known as elements. A subset would be a selection of these elements. The elements of a set could be a set of integers, shapes, people etc.
Sets. A set is a collection of things. For example, the items you wear is a set: these include hat, shirt, jacket, pants, and so on. You write sets inside curly brackets like this: {hat, shirt, jacket, pants, ...} You can also have sets of numbers: Set of whole numbers: {0, 1, 2, 3, ...}
Apr 8, 2012 · We often identify the powerset P(X) P (X) with the set of functions 2X 2 X, since we can think of the latter set as the set of characteristic functions of subsets of X X. Let f ∈2X f ∈ 2 X be a function and let Z ⊆ X Z ⊆ X. We can stipulate that if f(x) = 1 f (x) = 1 then x ∈ Z x ∈ Z, and if f(x) = 0 f (x) = 0 then x ∉ Z x ∉ Z.
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The symbol × × is used to denote the "Cartesian Product" of two sets: it results in a set with ordered pairs. The Cartesian product (some call it the "cross product" of sets) X × Y X × Y is defined such that. X × Y = {(x, y) ∣ x ∈ X, y ∈ Y} X × Y = {(x, y) ∣ x ∈ X, y ∈ Y} I'll get you started:
