This problem is also confusing me, can anyone shed some light on at least how to expand one of them?
For each of the following:
i. Write/expand each of the following predicates using disjunctions, conjunctions, and negations.
ii. Determine the truth value of the following statements.
Given:
P(x,y): x < y^3
Q(x,y): x –1 > y^2
R(x,y,z): x + y^3 > z
Universe of discourse for variable x is N (natural numbers)
Universe of discourse for variable y is Z+ (positive integers)
Universe of discourse for variable z is R+ (positive real numbers)
a.
b.
c.
d.
I understand the problem except that in the discrete book the problems always give a set of particular numbers to use the conjunctions, disjunctions, and negations on.
Here's what I'm going to do. Since the universe of discourse is infinite, it is impossible to exhaustively write/expand the predicates. I chose to write the first few in the sequence, then use ellipses to infer the existence of the rest.
If you have any other ideas, let me know
So, you'd go
(1 < 2 ^ 3) & (2 < 2 ^3) & (3 < 2 ^ 3) & (4 < 2 ^ 3) & (5 < 2 ^ 3) & (6 < 2 ^ 3) & (7 < 2 ^ 3) & (8 < 3 ^3).....and so on for
VxEy(x < y ^ 3) where x is a natural number and y is an integer greater than zero?
& = conjunction, < = less than, ^ = power.
I wish I knew how to formulate < and ^ in predicate logic instead of using mathematical symbols....
I'm not in your class. I'm just an interested bystander.
I was thinking the same.
For the expansion, I was thinking more of something along the lines of
(P(1,1) V P(1,2) V P(1,3) V …) ^ (P(2,1) V P(2,2) V P(2,3) V …) ^ (P(3,1) V P(3,2) V P(3,3) V …) ^ ...
I would use this for part ii, to show how I arrived at the predicate being true.
What do you guys think?
That's exactly how I did mine, except in a matrix form:
[P(0,1) v P(0,2) v P(0,3)] ^ ...
[P(1,1) v P(1,2) v P(1,3)] ^ ...
[P(2,1) v P(2,2) v P(2,3)] ^ ...
.
.
.
Remember that x's domain is natural numbers, and y's is positive integers (z is not bound, so I disregarded it).