2.1 Set

  • Element, Empty set (∅={}\emptyset = \{\})

    Prove ∅⊆A\emptyset \subseteq A.
    Proof:
    We need prove ∀x(x∈∅→x∈A)\forall x(x\in \emptyset \to x\in A). Since the assumption x∈∅x\in \emptyset is false, the conculsiont x∈Ax \in A is always true. Thus ∀x(x∈∅→x∈A)\forall x(x\in \emptyset \to x\in A) is ture.

  • Subset (⊂\subset and ⊆\subseteq) - A⊆B∀x(x∈A→x∈BA \subseteq B \quad \forall x (x \in A \to x \in B)
  • Powerset - the set of all subsets of a set. The number of subsets of a set with nn elements is 2n2^n.
  • Cartesian Products - A×B={(a,b)∣a∈A∧b∈B}A\times B = \{ (a, b) \mid a\in A \land b\in B\}