Quantifiers.md (1457B)
1 # Quantifiers 2 3 U 1.4.2 4 5 **Definition:** Quantifiers are operators that describe the number of individuals in a domain that satisfy something. 6 7 The two common quantifiers are: 8 9 1. $\exists$ - There exists (existential quantifier) 10 2. $\forall$ - For all (universal quantifier) 11 12 Another derived one is $\exists !$ which means there exists only one. Another way to state this is $\exists_1$. By using this notation we can then specify there are only an arbitrary number of elements of the set that have some property. 13 14 ### Propositional Functions 15 16 Quantifiers can be used to turn a propositional function into a proposition much like numbers. 17 18 $\forall x P(x)$ - We would need to specify some universe, but this is a proposition and not a propositional function 19 20 $P(x)$ - This is a propositional function 21 22 $P(1)$ - This is a proposition 23 24 ### Negation 25 26 When negating a for each we negate the propositional function (not P(x)) and flip the for each to be there exists. 27 28 The opposite is also true for negating a there exists. 29 30 Examples: 31 32 For all - $\neg \forall x P(x) = \exists x \neg P(x)$ 33 34 There exists - $\neg \exists P(x) = \forall x \neg P(x)$ 35 36 ### Scope 37 38 The scope of a quantifier is limited to what is contained in the parenthesis or preceeding propositional function when of the form $\forall x P(x)$. As such, $\forall x P(x) \to C(x)$ is not defined as we would need to specify $\forall x (P(x) \to C(x))$. This avoids an ambiguity in our statements.