MAT102 Lecture 05B

• 1
  • $\mathrm{\forall }q\in \mathbb{Q},\mathrm{\exists }r\in \mathbb{Q},(q>0)⟹0<r<q$
    • For every $q\in Q$ there is an $r\in Q$ such that if $q>0$, then $0<r<q$
    • For every rational $q$, if $q$ is positive, there is a smaller rational $r$ such that $r$ is greater than $0$ yet smaller than $q$.
    • There is no smallest positive rational number
  • $\mathrm{\forall }q\in \mathbb{Q},\mathrm{\forall }r\in \mathbb{Q},(q<r)\vee (r<q)$
    • This is false, because it's not two distinct numbers.
  • $\mathrm{\exists }q\in \mathbb{Q},\mathrm{\forall }r\in \mathbb{Q},0<q<r$
    • This is a smallest positive rational number
  • $\mathrm{\exists }q\in \mathbb{Q},\mathrm{\exists }r\in \mathbb{Q},(q<r)\wedge (r>0)$
    • There are two positive rationals where one is positive, and one is less than the other.
• 2
  • $\mathrm{\exists }d>0,\mathrm{\forall }r\in \mathbb{R},\mathrm{\exists }q\in \mathbb{Q},|r-q|<d$
    • There is a distance such that every real number has one rational
    • There is a precision such that every real number can be well approximated by a rational, accurate to that precision
  • $\mathrm{\forall }d>0,\mathrm{\forall }r\in \mathbb{R},\mathrm{\exists }q\in \mathbb{Q},|r-q|<d$
    • For any distance and real number, there is a rational
    • Correct
    • This implies the above statement, because if there is any distance, then it must work for one distance.
  • $\mathrm{\exists }q\in \mathbb{Q},\mathrm{\forall }d>0,\mathrm{\forall }r\in \mathbb{R},|r-q|<d$
    • There is a rational where for every distance you can approximate any real number
    • Every real number is able to be approximated by $q$
    • Suppose such a $q$ exists:
      • show two different real numbers, show it violates distance
  • $\mathrm{\forall }r\in \mathbb{R},\mathrm{\exists }q\in \mathbb{Q},\mathrm{\forall }d>0,|r-q|<d$
    • For every real number there is a rational where all distances can approximate it
    • This says every real number is rational
    • It forces the $q=r$ because $q$ must work for every $d$
    • Counterexample:
      • Find a real, show that there is no rational that works for every distance
• 6
  • $\mathrm{\forall }e\in \mathrm{\varnothing }$ is even
  • $2k$
  • $\{0\}$
  • $\{\}$

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  • $\mathrm{\exists }x\in \mathrm{\varnothing },$ x is even: false
  • There is no even number, because there's nothing
  • $\mathrm{\exists }x\in \mathrm{\varnothing },$ x is odd: false
  • $\mathrm{\exists }x\in \mathrm{\varnothing },P(x)$: false
  • Exists, requires that something be in the set
  • Negate it, then it becomes true.
  • $\mathrm{\neg }(\mathrm{\exists }x\in \mathrm{\varnothing },P(x))$
  • $\mathrm{\forall }x\in \mathrm{\varnothing },\mathrm{\neg }P(x)$: Always true
  • If it's forall, then it's always true
  • If $P(x)$ then $Q(x)$
    • $\mathrm{\forall }x\in \{z:P(z)\},Q(x)$
    • Vacuous truth