Nested Quantifiers

1. Nested Quantifiers

2. Nested Quantifiers • Needed to express statements with multiple variables • Example 1: “x+y = y+x for all real numbers” • xy(x+y = y+x) • where the domains of x and y are real numbers • Example 2: “Every real number has an inverse” • x y(x + y = 0) • where the domains of x and y are real numbers • Each quantifier is within the scope of the preceding quantifier: • x y(x+y=0) can be viewed as • x Q(x) where Q(x) is y P(x, y) where P(x, y) is (x+y=0)

3. Order of Quantifiers Example 1: • Let P(x,y) be the statement “x+y=y+x” • Assume that U are real numbers • Then xyP(x,y)≡yxP(x,y)≡Tbecause for any combination of x and y the statement holds Example 2: • Let Q(x,y) be the statement “x+y=0” • Assume that U are real numbers. • Then xyP(x,y)≡T, because for every x there is always an inverse y • butyxP(x,y)≡F because for a given y not every x will add up to zero

4. Quantifications of Two Variables • General rules:

5. Translating Math Statements Example : “The sum of two positive integers is always positive” Solution: • Rewrite the statement to make the implied quantifiers and domains explicit: • “For every two positive integers, their sum is positive.” • Introduce variablesx and y: • “For all positive integers x and y, x+y is positive.” • Introduce quantifiers: • xy ((x > 0)∧ (y > 0)→ (x + y > 0)) • where the domain of both variables is integers

6. Translation from English • Decide on the domain; choose the obvious predicates; express in predicate logic. • Example 1: “Brothers are siblings.” • domain: all people • x y (B(x,y) → S(x,y)) • Example 2: “Siblinghood is symmetric.” • x y (S(x,y) → S(y,x)) • Example 3: “Everybody loves somebody.” • x yL(x,y)

7. Translation from English • Example 4: “There is someone who is loved by everyone.” • yxL(x,y) • Example 5: “There is someone who loves someone.” • xyL(x,y) • Example 6: “Everyone loves himself” • xL(x,x)

8. Negating Nested Quantifiers • Recall De Morgan’s laws for quantifiers • ¬x P(x) ≡ x ¬P(x) • ¬x P(x) ≡ x ¬P(x) • Starting from left to right, successively apply these laws to nested quantifiers • Example: ¬w a f (P(w,f ) ∧ Q(f,a)) ≡ ≡ w ¬a f (P(w,f ) ∧ Q(f,a)) by De Morgan’s for  ≡ w a ¬f (P(w,f ) ∧ Q(f,a)) by De Morgan’s for  ≡ w a f ¬(P(w,f ) ∧ Q(f,a)) by De Morgan’s for  ≡ w a f (¬P(w,f ) ∨¬Q(f,a)) by De Morgan’s for ∧