1 - Sets - 1.1, 3.1

ucla | MATH 61 | 2022-09-23T11:05

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Table of Contents

• Recall
• Notes
• Lecture

Recall

1. Sets are a collection of unique elements
   1. cardinality is the quantity of a set
2. We can operate on sets using
   1. unions/intersections
   2. differences/complements
   3. collections/partitions
   4. products
3. We can describe set correlations as
   1. injective, surjective, bijective
4. Functions we use
   1. mapping/transformations, modulus, floor/ceiling, composition

Notes

Core Definitions - 1.1

• sets
  • collection of objects called elements (elts) or members
  • if few element
    • $A=1,2,3$
  • many elements
    • $B=1,2,3,\dots$
  • conditional
    • $C=2,4,6,8,\dots =x\in \mathbb{Z}|\text{x is an even integer}$
• symbols

  • $\mathbb{Z}={x

    x\space\text{is an integer}}$

  • $\mathbb{Q}={x

    x\space\text{is a rational number}}$

  • $\mathbb{R}={x

    x\space\text{is a real number}}$

• infinite/finite
  • if x has infinite elements, x is infinite else finite
• cardinality
  • the number of unique elements in x written $|x|$
• empty set
  • if x contains no elements, denote empty set as $\varphi$
• equal sets
  • if each set contains the same elements
  • prove equality by showing “$\text{if}x\in X\text{then}x\in Y\text{and}\text{if}y\in Y\text{then}y\in X$
  • prove inequality by showing $\text{some}x\in X\text{where}x\notin Y$
• subset
  • if each element of X is also an element of Y, X is a subset of Y, denoted $X\subseteq Y$else $X⊈Y$
• proper subset
  • if $X\subseteq Y$ and $X\ne Y$ then $X\subset Y$
• power set
  • the set of all subsets of X denoted $P(X)$
  • if $X=1,2$ then $P(X)=\varphi ,1,2,1,2$
• universal set
  • the usual larger set in which all sets are contained denoted $U$

Set Operations - 1.1

• union
  • $X\cup Y=z|z\in X\text{or}z\in Y$
• intersection
  • $X\cap Y=z|z\in X\text{and}z\in Y$
  • if $X\cap Y=\varphi$ then X and Y are disjoint
• difference
  • $X-Y=z|z\in X\text{and}z\notin Y$
• complement
  • $\overline{X}=x\in U|x\notin X=U-X$
• collection
  • set of sets denoted denoted
  • ${\cup }_i{A}_i=x\in A_i\text{for some}i$
  • ${\cap }_j{A}_j=x\in A_j\text{for each}j$
• partition
  • a collection $S=A_i$ is a partition of set X if for each $x\in X$ there is some $i$ for which $x\in A_i$ and each $A_i,A_j$ are disjoint
• cartesian product

  • $X\times Y={(x,y)

    x\in X,y\in Y}$

Functions - 3.1

• function
  • assignment of an element of Y (codomain) to each of X (domain)
  • $f:X\to Y$ and $f(x)$

  • the range is ${y\in Y

    y=f(x)\space\text{for some} x\in X}$

• modulus
  • x mod y is the remainder from dividing x by y
• floor
  • round down to nearest integer denoted $\lfloor x\rfloor$
• ceiling
  • round up to nearest integer denoted $\lceil x\rceil$
• injective (one-to-one)
  • $f:X\to Y$ is injective if:
  • for each $x_1,x_2\in X$ if $f(x_1)=f(x_2)$ then $x_1=x_2$
• surjective (onto)
  • $f:X\to Y$ is surjective if:
  • for each $y\in Y$ there exists some $x\in X$where $f(x)=y$
• bijective (invertible)
  • $f:X\to Y$ is bijective if:
  • $f$ is injective and surjective, then:
  • $f^{-1}$ is the inverse of $f:X\to Y$ where if $f(x)=y$ then $f^{-1}(y)=x$ denoted $f^{-1}:Y\to X$
• composition
  • for $f:X\to Y,g:Y\to Z$:
  • $g\circ f=g(f)$ such that $g\circ f(x)=g(f(x))=g(y)=z$

Lecture

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**SUMMARY
**In Math 61, we explore sets - a mathematical object used to store elements; elements can be of any type including other sets which are called subsets
We explore manipulating sets using set operations and functions to describe or transform sets