MAT102 Lecture 23

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  • My answers are correct if these are finite sets, or else you can make weird sets like $(0,1)\to \mathbb{R}$ which have the same cardinality.

• If we have $\mathbb{R}\setminus \mathbb{Q}$ it's uncountably infinite:

  • $\mathbb{R}$ is uncountable
  • $\mathbb{Q}$ is countably infinite

Worksheet 12

• 1
  • a
    • Let's show that $|A|\cdot |A|=|A\times A|=2|A|$. This is how it would work with non-negative integers.
    • Let $A=\{1,2\}$
    • $|\{1,2\}\times \{1,2\}|=2|\{1,2\}|$
    • $|\{(1,1),(1,2),(2,1),(2,2)\}|=2|\{1,2\}|$
    • $4=2|\{1,2\}|$
    • $4=2(2)$
    • $4=4$
    • in this example it works.
  • b
    • Addition:
      • Commutative:
        • We know that if we add the number of elements int two sets, it doesn't matter if we add the elements from set $A$ to set $B$ or vice versa. We will still be adding the same amount of elements.
      • Associative:
        • We know that it doesn't matter what order we add the number of elements in a set. So if we add the number of elements from $A$ and $B$, then add $C$, it won't change if we add $B$ and $C$ then $A$.
    • Multiplication:
      • If we abstract away the $||$ from the sets. So we can just say that $|A|=a$ and $|B|=b$. If $a,b\in \mathbb{R}$, then it should inherit field axioms from $\mathbb{R}$, such as commutativity and associativity. So this should hold for multiplication.
  • c
    • Suppose $a,b$ are disjoint sets. Without loss of generality.
    • Show that if $|A|\le |C|$ and $|B|\le |D|$, then $|A|+|B|\le |C|+|D|$
    • $|A|+|B|=|A\cup B|$ we can do this because we assumed $a,b$ are disjoint sets.

Solutions

• 1
  • c
    • Assume $\{\begin{array}{l}f:A\to C\\ g:B\to D\end{array}$ are both injections.
    • Assume everything is disjoint.
    • We need an injection $h:A\cup B\to C\cup D$
    • Define $h(x)=\{\begin{array}{ll}f(x)& x\in A\\ g(x)& x\in B\end{array}$
      • * If $x\in A$ then $x$ should take on the $f(x)$ function to get $C$.
      • Suppose $h(x_1)=h(x_2)$
      • Case 1: both are in a
      • Case 2: both are in b
      • Case 3: $x_1\in A,x_2\in B$
  • d
    • Define $h:A\times B\to C\times D$
    • $h(a,b)=\{\begin{array}{l}(f(a),g(b))\end{array}$