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.
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.
The statement is true:
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
what is the law of syllogism?
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
Just pointing that out.