Sets and Cardinality #proofs

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All of mathematics can be described using sets. But what are sets? A set is a collection of things, and those things are called elements of the set. A set can be anything, but here we’ll focus on how they’re used in math, so our sets will consist of numbers, functions, points, etcetera. For example, {2, 4, 6, 8} is a set with four elements, the numbers 2, 4, 6, 8. Sets can also be infinite, like the set of integers {...,-1, 0, 1,...} the dots indicating that it goes on forever. A set that has infinitely many elements is called an infinite set. Otherwise it is called a finite set. And we say two sets are equal if they contain the exact same elements. So the sets {2, 4, 6, 8} and {4, 2, 8, 6} are equal, even though they’re listed in a different order, their individual elements are identical. But {2, 4, 6, 8} and {2, 4, 6, 7} are not equal since 7 is only in one set. And we’ll use capital letters to denote sets. So we’ll say A = {2, 4, 6, 8} and we’ll say 2 ∈ A to denote that 2 is an element of A. 2, 4, 6, 8 ∈ A, but 5 !∈ A denotes 5 is not in A. There’s a few sets that come up so frequently they have designated symbols. N for the set of natural numbers. N is {1, 2, 3, …}, Z is the set of integers {..., -3, -2, -1, 0, 1, 2, 3, …}, and R being the set of real numbers. Sets can be other things besides numbers. We can have B = {T, F} where the values are true and false, C = {a, e, i, o, u} the set of lowercase vowels in english, or D = {(0,0),(1,0),(0,1),(1,1)} being a set of four points in R2. You can also have sets inside sets, for example E = {1, {2, 3}, {2, 4}} and we see 1 ∈ E, {2, 3} ∈ E, but 2, 3, 4 !∈ E. We also have the notion of size of a set, which we’ll call cardinality. It’s simply the number of elements in a set, and we’ll denote it as the set with vertical bars next to it. So |A| = 4, |B| = 2, |C| = 5, |D| = 4, and |E| = 3. We also have a special set called the empty set {} which contains no elements. And if we have a set containing I = {N, Z} we can think of it as a container containing two containers, the box of natural numbers, and the box of integers. So |I| = 2. We’ll use set builder notation to describe large sets. For example if we let E = {2n : n in Z} this is the set of even integers. The first brace reads “the set of all things of form”, and the colon is read “such that” So we read E as “the set of all things of form 2n, such that n is an element of Z”.

this video is in English

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