Math Help - Discrete Math problem

1. Discrete Math problem

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2. Originally Posted by chillerbros17
4. Let the domain D be the set of nonzero real numbers. D is the domain for both x and y. For each part, decide whether the statement is true or false. If statement is true, explain why. If statement is false, find negation and explain why negation is true.

a. (There exists x) (for all y) xy= 1
b. (For all x) (There exists y) xy=1
c. (For all x) (for all y) [ sin^2x + cos^2x = sin^2y + cos^2y ]
d. (There exists x) (There exists y) [ 2x+y = 5 ^ x-3y = -8 ]
a) False. Let y = 0. Then x does not exist.
b) False. Same argument as in a).
c) True. [sin(x)]^2 + [cos(x)]^2 = 1 for all x (and similarly for all y).
d) True. See attached graph.

3. Originally Posted by topsquark
a) False. Let y = 0. Then x does not exist.
b) False. Same argument as in a).
Since x and y are dreawn from the non-zero reals both of these are in fact true.

RonL

4. Originally Posted by topsquark
d) True. See attached graph.
False as the curves do not meet when =-8.

RonL

5. Originally Posted by CaptainBlack
Since x and y are dreawn from the non-zero reals both of these are in fact true.

RonL
(explicative) I hate it when I miss details like that.

-Dan

6. Originally Posted by CaptainBlack
False as the curves do not meet when =-8.

RonL
2x+y = 5 ^ x-3y = -8

2x + y = -8
y = -2x - 8

5^x - 3y = -8
3y = 5^x + 8

y = (1/3)*5^x + 8/3

which are the two functions I graphed.

-Dan

7. Originally Posted by chillerbros17
4. Let the domain D be the set of nonzero real numbers. D is the domain for both x and y. For each part, decide whether the statement is true or false. If statement is true, explain why. If statement is false, find negation and explain why negation is true.
a. (There exists x) (for all y) xy= 1
b. (For all x) (There exists y) xy=1
Proposition (a) is false. It says that there is a number in the domain that has the property that it times any other number in the domain equals 1: clearly false.
Its negation is (For all x)(there exists a y)[xy<>1].

Proposition (b) is of course true for the nonzero reals are an Abelian group.

8. What is the negation of (There exists x) (There exists y) [ 2x+y = 5 ^ x-3y = -8 ]

9. Originally Posted by chillerbros17
What is the negation of (There exists x) (There exists y) [ 2x+y = 5 ^ x-3y = -8 ]
[For all x][for all y]{2x+y<>5 OR x-3y<>-8} where "<>" means not equal.