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MATH0005 Algebra 1 part 1 - sets, functions, relations, and permutations. MATH0005 Algebra 1: sets, functions, and permutations. These are the lecture notes for the first part of MATH0005: Algebra 1. The notes are split into numbered sections, one for each teaching video. The notes are very similar to the video content, so you may find it ...
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Dec 26, 2022 · 2.1.1 Definition of a set A set is a collection of (mathematical) objects. There is an entire field of mathematics called set theory dedicated to the study of sets and to their use as a foundation for mathematics, but in MATH0005 we are going to give only an informal introduction to sets and their properties.
Theorem 1.1.1 1.1. 1. Two sets A A and B B are equal if and only if A ⊂ B A ⊂ B and B ⊂ A B ⊂ A. If A ⊂ B A ⊂ B and A A does not equal B B, we say that A A is a proper subset of B B, and write A ⊊ B A ⊊ B. The set θ = {x: x ≠ x} θ = {x: x ≠ x} is called the empty set. This set clearly has no elements.
- Definition
- Notation
- Numerical Sets
- Why Are Sets Important?
- Some More Notation
- Equality
- Subsets
- Proper Subsets
- Even More Notation
- Empty (or null) Set
What is a set? Well, simply put, it's a collection. First we specify a common property among "things" (we define this word later) and then we gather up all the "things" that have this common property. For example, the items you wear: hat, shirt, jacket, pants, and so on. I'm sure you could come up with at least a hundred. This is known as a set. So...
There is a fairly simple notation for sets. We simply list each element (or "member") separated by a comma, and then put some curly brackets around the whole thing: This is the notation for the two previous examples: {socks, shoes, watches, shirts, ...} {index, middle, ring, pinky} Notice how the first example has the "..." (three dots together). S...
So what does this have to do with mathematics? When we define a set, all we have to specify is a common characteristic. Who says we can't do so with numbers? And so on. We can come up with all different types of sets. We can also define a set by its properties, such as {x|x>0} which means "the set of all x's, such that x is greater than 0", see Set...
Sets are the fundamental property of mathematics. Now as a word of warning, sets, by themselves, seem pretty pointless. But it's only when we apply sets in different situations do they become the powerful building block of mathematics that they are. Math can get amazingly complicated quite fast. Graph Theory, Abstract Algebra, Real Analysis, Comple...
Now you don't have to listen to the standard, you can use something like m to represent a set without breaking any mathematical laws (watch out, you can get πyears in math jail for dividing by 0), but this notation is pretty nice and easy to follow, so why not? Also, when we say an element a is in a set A, we use the symbol to show it. And if somet...
Two sets are equal if they have precisely the same members. Now, at first glance they may not seem equal, so we may have to examine them closely! And the equals sign (=) is used to show equality, so we write: A = B
When we define a set, if we take pieces of that set, we can form what is called a subset. In general: So let's use this definition in some examples. Let's try a harder example.
If we look at the defintion of subsets and let our mind wander a bit, we come to a weird conclusion. Let A be a set. Is every element of A in A? Well, umm, yes of course, right? So that means that A is a subset of A. It is a subset of itself! This doesn't seem very proper, does it? If we want our subsets to be proper we introduce (what else but) pr...
When we say that A is a subset of B, we write A B. Or we can say that A is not a subset of B by A B ("A is not a subset of B") When we talk about proper subsets, we take out the line underneath and so it becomes A B or if we want to say the opposite, A B.
This is probably the weirdest thing about sets. As an example, think of the set of piano keys on a guitar. "But wait!" you say, "There are no piano keys on a guitar!" And right you are. It is a set with no elements. This is known as the Empty Set(or Null Set).There aren't any elements in it. Not one. Zero. It is represented by Or by {}(a set with n...
- Sets Definition. In mathematics, a set is defined as a well-defined collection of objects. Sets are named and represented using capital letters. In the set theory, the elements that a set comprises can be any kind of thing: people, letters of the alphabet, numbers, shapes, variables, etc.
- Representation of Sets in Set Theory. There are different set notations used for the representation of sets in set theory. They differ in the way in which the elements are listed.
- Sets Symbols. Set symbols are used to define the elements of a given set. The following table shows the set theory symbols and their meaning. Symbols. Meaning.
- Types of Sets. There are different types of sets in set theory. Some of these are singleton, finite, infinite, empty, etc. Singleton Sets. A set that has only one element is called a singleton set or also called a unit set.
Our resource for Algebra 1: An Incremental Development includes answers to chapter exercises, as well as detailed information to walk you through the process step by step. With Expert Solutions for thousands of practice problems, you can take the guesswork out of studying and move forward with confidence. Find step-by-step solutions and answers ...
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For us, a set will simply be an unordered collection of objects. Two examples: we could consider the set of all actors who have played The Doctor on Doctor Who, or the set of natural numbers between 1 and 10 inclusive. In the first case, Tom Baker is an element (or member) of the set, while Idris Elba, among many others, is not an element of ...
