Help me with the following.

2. Prove that the law of syllogism is a tautology.

8. If a divides b, then a <= b.

14. Prove or disaprove - The product of two odd integers is odd.

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- April 26th 2007, 12:08 AMj-downDiscrete Math
Help me with the following.

2. Prove that the law of syllogism is a tautology.

8. If a divides b, then a <= b.

14. Prove or disaprove - The product of two odd integers is odd. - April 26th 2007, 12:30 AMecMathGeek
8 and 14 I can do.

8: If a | b, where a and b are positive whole numbers then b = ak for k is some positive whole number: k = 0, 1, 2, 3.... Therefore, a <= b.

14: Since the form of an odd number is 2n + 1 for n = 0, 1, 2...

Let 2a + 1 and 2b + 1 be two odd integers

(2a + 1)(2b + 1) = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1 which is an odd integer. - April 26th 2007, 12:30 AMJhevon
here's 14:

The statement is true:

Proof

let a and b be two odd integers. then a = 2n + 1 and b = 2m + 1 for some integers m and n.

so a*b = (2n + 1)(2m + 1) = 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1 = 2c + 1, where c = 2mn + m + n is an integer

so we see a*b is odd

QED

what is the law of syllogism? - April 26th 2007, 12:35 AMecMathGeek
- April 26th 2007, 12:48 AMJhevon
Okay, so if my memory serves me right, the law of syllogism goes like this:

for statements P, Q and R

if (P=>Q) and (Q=> R) then (P=>R)

the proof is done using truth tables, see the image below. notice that the last column has all true values. a statement that is true in every instance is called a taughtology (of course i don't remember if this is the exact law of syllogism, but whatever it is, it's a taughtology. so it will at least give you an idea for the solution - April 26th 2007, 06:21 AMThePerfectHacker
Number 8 is not true.

2 divides -2 but yet,

2<=-2 :eek:

You should have used absolute values. - April 26th 2007, 07:29 AMecMathGeek
- April 26th 2007, 08:01 AMtopsquark
- April 26th 2007, 08:40 AMecMathGeek
- April 26th 2007, 09:49 AMtopsquark
Not

*completely*irrelevant. Though it is very likely that we are using the real number system, given that the class is Discrete Math it is not outside of the realm of reason to use modular Mathematics, in which case the theorem doesn't hold.

Just pointing that out. :)

-Dan