2.7 Quantifiers

Class: CSE 16 Subject: computer-science discrete-math Date: 2024-10-09 Teacher: Prof. Musacchio

Logic

Quantifier Symbols

1. For All

• for all this is represented by the symbol ∀
  • E.g. “Every element of $X$ is odd.” ⇒ $\forall x\in X,\mathbb{P}(x)$
• The format above is referred to as the universal quantification

2. There Exists

• there exists is represented by the symbol ∃
  • E.g. “There is at least one element of $X$ that is odd,” $\exists x\in X,\mathbb{P}(x)$
• The format above is referred to as the existential quantification

Nested Quantifiers

• E.g. $S_n$ , $n\in \mathbb{N}$ :
  • $S_n$ = $1/n$ $\forall ϵ\in (0,\infty ),$ there exists n in N, for all m >= n| |Sn - 0| < epsilon ⇒ True

Negation

• there exists epsilon in (0, infinity), ~ (there exists n in N, for all m >=n, |Sn - 0| < epsilon)
• there exists epsilon in (0, infinity), for all n in N, ~(or all m >=n, |Sn - 0| < epsilon)
• there exists epsilon in (0, infinity), for all n in N, there exists m >= n, |Sn - 0| >= epsilon