1. Discrete Mathematical الرياضيات المتقطعة

2. Example 12 June 20162 OR Q(x,y): x+y=x-y a) Q(1,1): 2=0 False b) Q(2,0): 2+0=2-0 True c) Q(1,y): 1+y=1-y False(take any y<>0, x: y=1) d) Q(x,2): x+2=x-2 False

3. First: SOLUTION Q1. (5 pts) Show that the following argument form is invalid: 6/12/2016

5. Q2. Use the truth table to show if the argument is valid. " If this number is larger than 2, then its square is larger than 4." " This number is not larger than 2. " The square of this number is not larger than 4. p → q  p  q 6/12/2016

6. Predicates - multiple quantifiers (Nested quantifiers) To bind many variables, use many quantifiers! Example: P(x,y) = “x > y”  x P(x,y)  x  y P(x,y)  x  y P(x,y)  x P(x,3) a)True proposition b)False proposition c)Not a proposition d)No clue c)b)a)b)

7. 6/12/2016 Predicates - the meaning of multiple quantifiers  x  y P(x,y)  x  y P(x,y)  x  y P(x,y)  x  y P(x,y) P(x,y) true for all x, y pairs. For every value of x we can find a (possibly different) y so that P(x,y) is true. P(x,y) true for at least one x, y pair. There is at least one x for which P(x,y) is always true. quantification order is not commutative.

8. 6/12/2016 Predicates - the meaning of multiple quantifiers N(x,y) = “x is sitting by y”  x  y N(x,y)  x  y N(x,y)  x  y N(x,y)  x  y N(x,y) False True? True False

9. 6/12/2016 Multiple quantifiers (Examples)  x  y, P(x,y): For all x and for all y the relation P(x,y) is true. If two numbers are integers then their product is an integer. 2.  x  y, P(x,y): For all x there is some y such that P(x,y) is true. Every student has a favorite teacher Note: here and below in all examples concerning people, we shall assume that the domain is known and will not represent it neither separately, nor within the predicate expression.

10. 6/12/2016 Multiple quantifiers (Examples) 3.  x  y, P(x,y): There is some x such that for all individuals y the relation P(x,y) is true. Someone is loved by everybody  x  y loves (y,x) There is a professor that is liked by all students 4.  x  y, P(x,y): There is some x and there is some y such that P(x,y) is true. Some students have favorite teachers

11. 11 Extra exmples for multiple quantifiers  x  y P(x, y) –“For all x, there exists a y such that P(x,y)” –Example:  x  y (x+y == 0)  x  y P(x,y) –There exists an x such that for all y P(x,y) is true” –Example:  x  y (x*y == 0)

12. 12 Order of quantifiers  x  y and  x  y are not equivalent!  x  y P(x,y) –P(x,y) = (x+y == 0) is false  x  y P(x,y) –P(x,y) = (x+y == 0) is true