Hi
Yeah your idea seems good, using existence quantifiers it becomes:
In
I have to express this using quantifiers, divisibility, = <= <, etc. “If a natural number is odd, then all its divisors are odd.”
I was thinking of, for all 2x+1, where x is a natural number, if 2x+1 is divisible by some value y, then y=2n+1 where n is a natural number. Not sure how to put it into a predicate logic and what not. Any help?
Predicates
N(x) : x is a natural number.
O(x) : x is an odd number.
D(y,x) : y divides x
Div(y,x) : y is a divisor of x
.
(I used an outer bracket [] rather than () for readibility).
Now, check several cases whether the above one is valid or not.
case 1. x is a not a natural number (vacuously true)
case 2. x is a natural number and x is not odd (vacuously true).
case 3. x is a natural number and y is not a divisor of x or y is not a natural number (vacuously true).
case 3. x is an odd natural number, y is a divisor of x and is not an odd natural number. (False)
case 4. x is an odd natural number, y is a divisor of x and y does not divide x. (False)
case 5. x is an odd natural number, y is a divisor of x and y is an odd natural number which divides x. (True)
Hello PlatoSorry, but you leave me still confused. In your posting where you referred to an order of precedence, you said it was
and in this post, you say that this reads from left to right, highest to lowest.
And yet your example would appear to demonstrate the opposite: namely that the highest precedence (i.e the operation that is carried out first) is given to , then , and so on. Thus your interpretation of is exactly the same as mine.
So are we talking at cross-purposes here, and we mean different things by 'highest to lowest'?
Grandad
I think that is quite possibly the case. Perhaps I should have said that the order is in terms of the scope of the operator. Therefore, has the least scope and has the greatest scope. I can direct you to Symbolic Logic by I.M.Copi. He has an extensive discussion of this topic. The example I gave is copied directly out of that textbook.