Statements Containing Multiple Quantifiers Lecture 9 Section 2

Statements Containing Multiple Quantifiers Lecture 9 Section 2. 3 Wed, Jan 26, 2005

Multiply Quantified Statements ¢ Multiple universal statements x S, y T, P(x, y) l y T, x S, P(x, y) l The order does not matter. l ¢ Multiple existential statements x S, y T, P(x, y) l y T, x S, P(x, y) l The order does not matter. l

Multiply Quantified Statements ¢ Mixed universal and existential statements x S, y T, P(x, y) l y T, x S, P(x, y) l The order does matter. l What is the difference? l ¢ Compare x R, y R, x + y = 0. l y R, x R, x + y = 0. l

Examples ¢ Which of the following statements are true? x N, y N, y < x. l x Q, y Q, y < x. l x R, y R, y < x. l x Q, y Q, z Q, x < z < y. l ¢ For those that are false, what is their negation?

Negation of Multiply Quantified Statements Negate the statement x R, y R, z R, x + y + z = 0. ¢ ( x R, y R, z R, x + y + z = 0) x R, ( y R, z R, x + y + z = 0) x R, y R, ( z R, x + y + z = 0) x R, y R, z R, (x + y + z = 0) x R, y R, z R, x + y + z 0 ¢

Multiply Quantified Statements ¢ ¢ In the statement x R, y R, z R, x + y + z 0 the predicate x+y+z 0 must be true for every y and for some x and for some z. However, the choice of x must not depend on y, while the choice of z may depend on y.

Negation of Multiply Quantified Statements ¢ Consider the statement n N, r, s, t N, n = r 2 + s 2 + t 2. ¢ Its negation is l n N, r, s, t N, n r 2 + s 2 + t 2. Which statement is true? ¢ How would you prove it? ¢

Example In the Theory of Computing there is an important theorem: The Pumping Lemma. ¢ The Pumping Lemma: For every regular set S, there exists a positive integer n, such that for every word w S for which |w| n, there exist words x, y, z, such that for every nonnegative integer k, xykz S. ¢

Example ¢ The Pumping Lemma: regular set S, a positive integer n, such that word w S for which |w| n, words x, y, z, such that nonnegative integer k, xykz S.

Example ¢ The Pumping Lemma: S R, n N, w S for which |w| n, x, y, z *, k N, xykz S.

Example ¢ The Pumping Lemma: S R, n N, w S for which |w| n, x, y, z *, k N, xykz S.

Example ¢ The Pumping Lemma: S R, n N, w S for which |w| n, x, y, z *, k N, xykz S.

Example ¢ The Pumping Lemma: S R, n N, w S for which |w| n, x, y, z *, k N, xykz S.

Example ¢ The Pumping Lemma: S R, n N, w S for which |w| n, x, y, z *, k N, xykz S.

Example ¢ The Pumping Lemma: S R, n N, w S for which |w| n, x, y, z *, k N, xykz S.