MATH2020 (Lec 01) - Set Theory

Set Theory

set definition

• unordered collection of distinct objects
  • eg. $V=\{u,e,o,a,i\}$
• set builder notation: $P=x\in {\mathbb{Z}}^{+}|xisprimenumberandx<20$
• data types in CS is built upon the concept of a set

•   my_set = {1,2,"hi"}
    # set from iterable object
    my_set = set([1,2, "hi"])

•   Set<Integer> mySet = newHashSet<>();

Equal Set

• every element in 1st set is in 2nd set, and vice versa $\forall x(x\in A\leftrightarrow x\in B)$
• Remark 1: empty set / null set is special set with no element $\varnothing$
• Remark 2: set with 1 element: singleton set

Subset:

• A is a subset of B, and B is a superset of A ⇐> every element of A is also element of B
• (A is subset of set B: $A\subseteq B$) = (B is superset of set A: $B\supseteq A$)
• Theorem 4: For every set S, (i) $\varnothing \subseteq S$, (ii) $S\subseteq S$
• Remark 1: A is proper subset of B if (i) A is a subset of B (ii) $A\ne B$
• Remark 2: $A\subseteq B$ and $B\subseteq C$ ⇒ $A=B$

Power Set:

• power set of S is set of all subsets of set S, $\mathcal{P}(S)$

Cardinality:

• If have exactly n distinct elements in set S, n is nonnegative integer → S is a finite set, n is cardinality (number of elements) of S (denoted as $|S|$)
• $\mathcal{P}(\{0,1,2\})=\{\varnothing ,\{0\},\{1\},\{2\},\{0,1\},\{1,2\},\{0,2\},\{0,1,2\}\}$

Ordered n-tuples

• represent ordered collection - with tuple $(a_1,a_2,...,a_n)$
  • as this is ordered, $(1,2)\ne (2,1)$
  • $(a_1,a_2,...,a_n)=(b_1,b_2,...,b_n)$ ⇐> $a_i=b_i$ for all $i=1,2,...,n$

Cartesian Product

• A x B, is set of all ordered pairs (a, b), where $a\in A$ and $b\in B$ $A\times B=\{(a,b)|a\in A\wedge b\in B\}$
  • $\wedge$: and
• pairs every element of the first set with every element of the second set
• eg:
  • $A=\{1,2\},B=\{x,y\}$
  • $A\times B=\{(1,x),(1,y),(2,x),(2,y)\}$

Set operations

• Common set operations
  • Union: $S\cup T=\{x|x\in S\vee x\in T\}$: elements in S or T
    • $\vee$: or
  • Intersection: $S\cap T=\{x|x\in S,x\in T\}$: intersection of S and T only
  • Differences: $S-T=\{x|x\in S,x\notin T\}$: in S but not T
  • Complements: $\overline{S}=\{x|x\in \mathbb{U},x\notin S\}=\mathbb{U}-S$
    • $\mathbb{U}$ is the universe containing sets S and T
• Theorem 15
  • For all sets S and T, $S=(S\cap T)\cup (S-T)$
  • Prove Set Equality: show that $S\subset (S\cap T)\cup (S-T)$ and $(S\cap T)\cup (S-T)\subset S$
  • see the prove here

Strings

• Given a set $S$ and a natural number $n\in N$,
• $S^n$ is the set of length $n$ ‘strings’. Can be defined as product of $n$ copies of $S$ (i.e., $S\times S\times \cdots \times S$).”
  • $\{0,1{\}}^n$: set of all n-bit strings. $n=3$ → set is all possible combination of 0 and 1 in string of length 3
  • There are $2^n$ possible strings (8 when n = 3 above)
• $S^{\ast }$ is the set of finite length ‘strings’$S^{\ast }=S^0\cup S^1\cup S^2\cup ...$, where $S^0$ contains empty string.”
  • $\{0,1{\}}^{\ast }$: set of all finite-length bit strings. $\ast$ = any length, even 0. → set is all possible combinations of 0 and 1 for all possible lengths, including "", “0”, and so on
• $[n]$ is the set $\{0,1,2,\cdots ,n-1\}$

Russell’s Paradox

• $R=\{S|S\notin S\}$, that is R is the set of all sets that don’t contain themselves as an element
• see more here