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May 20, 2022 · \(\subseteq\): denotes that a set is a subset of another set. \(\subset\): denotes that a set is a proper subset of another set. \(\mid\): denotes for "such that" or "divides," depending on context. { } or \(\emptyset\): denotes an empty set. Equal sets: \(A = B\) if \(A \subseteq B\) and \(B \subseteq A\)
- 2.2: Operations With Sets
The empty set has just one subset, which is itself. The...
- 1.3: Subsets
Set A A is a subset of set B B if every member of set A A is...
- 9.3: Subsets and equality of sets
Proposition \(\PageIndex{1}\): Basic properties of the...
- 2.2: Operations With Sets
- 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...
\(A\) is a proper subset of \(B\) if \(A\) is a subset of \(B\) and \(A\) is not equal to \(B\). This is denoted by \( A \subset B \). The empty set \(\emptyset\) is a proper subset of every non-empty set. Subset versus proper subset: Take the following sets: \(A = \{a,b,c,d,e\}\), \(B = \{a,b,c,d,e\}\), and \(C = \{a,d,e\}\).
In set theory, a subset is denoted by the symbol ⊆ and read as ‘is a subset of’. Using this symbol we can express subsets as follows: A ⊆ B; which means Set A is a subset of Set B. Note: A subset can be equal to the set. That is, a subset can contain all the elements that are present in the set.
Set A A is a subset of set B B if every member of set A A is also a member of set B B. Symbolically, this relationship is written as A ⊆ B A ⊆ B . Sets can be related to each other in several different ways: they may not share any members in common, they may share some members in common, or they may share all members in common.
