Nesting Quantifiers And Their Manipulation Copyright © Curt Hill

Introduction The larger view is that a quantified expression is just a predicate –It must have a true or false value –Even if we do not know what it is Then it only makes sense that we may have a quantified expression within a quantified expression Copyright © Curt Hill

Commutativity Consider the statement of commutativity  i  j (i + j = j + i) –We have a second universal quantifier nested within the first The book prefers that notation, but I prefer:  i,j (i + j = j + i) –Where the quantifier has multiple dummies, all of which may range independently Copyright © Curt Hill

Associativity of Addition Use:  i  j  k (i + (j + k) = (i + j) + k) Or  i,j,k (i + (j + k) = (i + j) + k) Copyright © Curt Hill

Mixed The multiple quantifiers is required when they are different What does this mean?  i  j (i < j) How is this different from?  j  i (i < j) One of these is true and the other false Copyright © Curt Hill

Prime Definition Again This time with nested quantifiers prime(n) =  i    1 < i < n   j  I (j=n  i) Negation of an existential gives a universal so: prime(n) =  i    1 < i < n  j  I (j  n  i ) Copyright © Curt Hill

Scope Expression  i (pred1(i)   k pred2(i,k)) There are two bound variables i is known in entire range k is known only in second quantifier As far as the universal quantifier is concerned i is a free variable Both may be used in the second quantifier This may cause a problem if they use the same names Copyright © Curt Hill

Looping A quantifier has much in common with programming language loops with a few exceptions –They are often infinite –Since  and  are commutative the order does not matter This gives us a way to think about them Nested quantifiers then become nested loops Copyright © Curt Hill

Order If the two quantifiers are the same then we may reverse their order: (  i  j i + j = j + i) = (  j  i i + j = j + i) This is not generally true if there is a combination of universal and existential The inner quantifier is chosen after the outer  x (  y x+y = 0)   y (  x x+y = 0) –What do these two mean? –Which is true? Copyright © Curt Hill

Audience Participation For each of the following: –Is it true? –What does it mean? –Assume Reals if no type is given  x (  y x<y)  y (  x x<y)  i   (  j   i=j 2 )  i   (  j   j=i 2 ) Copyright © Curt Hill

Negation What happens when we negate an expression with nested quantifiers? We know how to negate a quantifier We just apply this from the outside in Recall  x (  y x+y = 0) This means that there exists a single real number that may be added to any real and the result is zero –We also know that this is false –Multiplication has a zero, not addition Copyright © Curt Hill

Negating  x (  y x+y = 0) Start with applying a single negation:   x (  y x+y = 0) Negation of an existential is a universal with negated predicate  x (   y x+y = 0) Negate again  x (  y x+y  0) Compare this with the more interesting:  y (  x x+y = 0) Copyright © Curt Hill

Summary We now know something about nesting quantifiers Translation these to and from English uses the same skills we used before and practice Interchanging nested quantifiers does not usually give us either the same or opposite results Copyright © Curt Hill

Exercises Rosen 1.5 Problems: 5, 9, 15, 21, 25, 31, 37, 43 Copyright © Curt Hill