Nested Quantifiers

2 Nested Iteration Let the domain be {1, 2, …, 10}. Let P(x, y) denote x > y.  x,  y, P(x, y) means  x, (  y, P(x, y) ) Is the above statement true?

3 Multiple Quantifiers  x,  y, P(x, y)  y,  x, P(x, y)  x,  y, P(x, y)  y,  x, P(x, y)  y,  x, P(x, y)  x,  y, P(x, y)  y,  x, P(x, y)  x,  y, P(x, y) Legend: AB is valid

4 Translate to English Let the domain be the real numbers.  x,  y, (((x ≥ 0)  (y 0)) Is there something wrong with  x, (((x ≥ 0)  (  y, y 0))

5 Translate to Locigal Expression Let Q(x,y) denote “student x has been a contestant on quiz show y” The domain for x is all students at UCSB. The domain for y is all quiz shows on TV. Express as a logical expression –Every TV quiz show has had a student from UCSB as a contestant. –At least 2 students from UCSB have been contestants on Jeopardy.

6 Translations  y  x Q(x, y).  x 1  x 2 ( (x 1  x 2 )  Q(x 1, Jeopardy)  Q(x 2, Jeopardy) )

7 Negating Nested Quantifiers Negate  x  y (P(x, y)  Q(x, y)) so that only predicates are negated. 1.  x  y (P(x, y)  Q(x, y)). 2.  x  y (P(x, y)  Q(x, y)). 3.  x  y  (P(x, y)  Q(x, y)). 4.  x  y (  P(x, y)   Q(x, y)).

8 Characters    ≥ ≡                        