Question:

I

What is the largest real number less than 10?
1- there is no largest real number less than 10 or
2- 9
or
3- 9.999...

II

Given that not (a and b) implies c.What does
(not c) imply ?
1-a and b
2- (not a and b) or (a and not b)
3- (not a and b) or (a and not b)
4- a or b

III

What are the x and y intercepts of the equation
2x + 3y -6 =0 ?
1- x=3 and y=2
2- x=6 and y=-2
3- -3x + 2y =0

I hope you can help me
Thank-you,

Hi Diane,

I

9.1 is a real number less than 10 and graeter than 9, so can't be #2
9.999... (if this is the infinite repeating decimal)
Let x = 9.999....
Then 10x = 99.999...
So 9x = 10x - x = 99.999... - 9.999... = 90
Thus, x = 90/9 = 10
Therefore, can't be #3.
Hence, by a process of elimination, the answer is #1. (note: this answer also makes sense because, for any real number less than 10, we can always find a larger real number less than 10 - the real numbers are infinitely dense)

II

a and b means both a and b are true
so not (a and b) means either
a is false and b is true (*), or
a is true and b is false (**), or
a is false and b is false (***).
Thus, not c implies not [not (a and b)], which means (*) and (**) and (***) are false. In other words, a is true and b is true. Therefore, not c means a and b. Hence, the answer is #1
Another way to see this: "not" undoes "not"
That is, not (not p) = p for any proposition p
Therefore, not c = not [not (a and b)] = a and b.

III

x intercept: set y = 0
2x + 3(0) - 6 = 0
2x = 6
x = 3
y intercept: set x = 0
2(0) + 3y - 6 = 0
3y = 6
y = 2
Therefore, the answer is #1.

Paul