# Quesons on order of quaners example 2 let domain be

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: w we have: Lewis Carroll Example ༉  The ﬁrst two are premises and the third is the conclusion. “All lions are ﬁerce.” 2.  “Some lions do not drink coﬀee.” 3.  “Some ﬁerce creatures do not drink coﬀee.” 1.  ༉  Let P(x), Q(x), and R(x) be the propositional functions “x is a lion,” “x is ﬁerce,” and “x drinks coﬀee,” respectively. 1.  ∀x (P(x)→ Q(x)) 2.  ∃x (P(x) ∧ ¬R(x)) 3.  ∃x (Q(x) ∧ ¬R(x)) Nested Quan\$ﬁers ༉  Nested quantiﬁers are often necessary to express the meaning of sentences in English as well as important concepts in computer science and mathematics. Example: “Every real number has an inverse” is ∀x ∃y (x + y = 0) where the domains of x and y are the real numbers. ༉  We can also think of nested propositional functions: ∀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) Thinking of Nested Quan\$ﬁca\$on ༉  To evaluate ∀ x ∀ y P(x, y), loop through the values of x : ༉  At each step, loop through the values for y. ༉  If for some pair of x and y, P(x, y) is false, ∀x ∀y P(x, y) is false and both the outer and inner loop terminate. ༉  To evaluate ∀ x ∃ y P(x, y), loop through the values of x: ༉  ༉  At each step, loop through the values for y. If no y is found such that P(x, y) is true, the outer loop terminates as ∀x ∃y P(x, y) has been shown to be false. Order of Quan\$ﬁers Examples: 1.  Let P(x, y) be the statement “x + y = y + x.” Assume that domain is the real numbers. Then ∀x ∀y P(x, y) and ∀y ∀x P(x, y) have the same truth value. 2.  Let Q(x, y) be...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern