- #1
Or ##ab<0 \Longrightarrow (a < 0 \wedge b>0) \vee (a>0 \wedge b<0)##Math_QED said: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.askor said: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.askor said: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).fresh_42 said:OR allows only one to be true,
Yeah, that was a misleading typeset. I didn't meant the Boolean OR, I wanted to emphasize:Mark44 said: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.HallsofIvy said: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].
fresh_42 said:Or ##ab<0 \Longrightarrow (a < 0 \wedge b>0) \vee (a>0 \wedge b<0)##
askor said:Does above applies too when ab > 0?
PeroK said:What are the conditions on ##a## and ##b## if ##ab > 0##?
askor said:I don't know.
PeroK said: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.
askor said:Do you mean if ##ab > 0## then ##a < 0 \wedge b < 0## or ##a > 0 \wedge b > 0##?
PeroK said: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.
askor said:Do you mean it was correct?
PeroK said:Yes. But, if you don't understand that notation properly I wouldn't use it.
askor said:The "##\wedge##" notation you mentioned about, it mean "and" isn't it?
PeroK said:Yes. And the other one means "or".
askor said:Which one is "or"?
As far as I know, it's too simple to have a name.askor said:What is the name of this inequality?
The sign changes from negative to positive when the value of a number changes from a negative number to a positive number. This is because the sign indicates the direction or orientation of the number on a number line. When the number changes from being on the left side of the number line (negative) to the right side (positive), the sign changes accordingly.
The sign switches from negative to positive when there is a change in the value of the number. This can happen when performing mathematical operations such as addition, subtraction, multiplication, or division. For example, when adding a negative number to a positive number, the result will be a positive number and the sign will change from negative to positive.
Yes, the sign can change from positive to negative. This happens when the value of a number changes from a positive number to a negative number. Similar to the previous question, this can occur when performing mathematical operations such as subtraction or division.
Yes, there is a specific rule for when the sign changes from negative to positive. This rule is based on the properties of numbers and mathematical operations. For example, when multiplying or dividing two numbers with different signs, the result will always be a negative number and the sign will change accordingly.
The sign change does not affect the value of the number itself, but it does affect its representation on a number line. The absolute value of the number remains the same, but the sign determines its direction on the number line. For example, the absolute value of -5 and 5 is both 5, but the sign indicates that -5 is on the left side of the number line while 5 is on the right side.