- #1
- 169
- 9
Or ##ab<0 \Longrightarrow (a < 0 \wedge b>0) \vee (a>0 \wedge b<0)##This just applies the following:
##a>0,b<0\implies ab<0##
In high school I learned that the product of two negative numbers is positive. And the product of a negative and a positive number is negative.Thank you for all of your answers but I still don't understand. I even can't find it in my algebra textbook. In what book is this kind of inequality is taught?
OR allows only one to be true, so ##(1 - \text{cos} ~x) > 0 ~\vee~ (\text{cos} ~x - \text{sin} ~x) < 0## is true in case ##(1 - \text{cos} ~x) > 0 ~\wedge~ (\text{cos} ~x - \text{sin} ~x) > 0##, whereas ##(1 - \text{cos} ~x)(\text{cos} ~x - \text{sin} ~x) < 0## is false.Why not like this?
##(1 - \text{cos} ~x)(\text{cos} ~x - \text{sin} ~x) < 0##
##(1 - \text{cos} ~x) > 0 ~\vee~ (\text{cos} ~x - \text{sin} ~x) < 0##
No, either or both operands can be true. Possibly you're thinking of "exclusive or" (XOR).OR allows only one to be true,
Yeah, that was a misleading typeset. I didn't meant the Boolean OR, I wanted to emphasize:No, either or both operands can be true. Possibly you're thinking of "exclusive or" (XOR).
You didn't mean that. Those two cases are the same.either
a) [itex]`1- cos(x)> 0[/itex] and [itex]cos(x)- sin(x)> 0[/itex]
or
b)[itex]cos(x)- sin(x)> 0[/itex] and [itex]`1- cos(x)> 0[/itex].
Or ##ab<0 \Longrightarrow (a < 0 \wedge b>0) \vee (a>0 \wedge b<0)##
Does above applies too when ab > 0?
What are the conditions on ##a## and ##b## if ##ab > 0##?
I don't know.
Well, that's an honest answer. Can you think of a way to find out?
You could multiply some numbers together and note whether the product is postive or negative. Then you could look for a pattern in the cases where the product is positive.
Do you mean if ##ab > 0## then ##a < 0 \wedge b < 0## or ##a > 0 \wedge b > 0##?
Personally I'd be happy to say that ##ab > 0## if both ##a## and ##b## are positive or both ##a## and ##b## are negative. Which is what you've written in "wedge" notation.
Do you mean it was correct?
Yes. But, if you don't understand that notation properly I wouldn't use it.
The "##\wedge##" notation you mentioned about, it mean "and" isn't it?
Yes. And the other one means "or".
As far as I know, it's too simple to have a name.What is the name of this inequality?