The first question is answered by Euclid's first proof. The answer is 120 degrees. The second question I can't answer. I haven't studied group theory yet so you'll have to fill me in. The last question is simple. If f(x)=f(1-x), then f'(x)=-f'(1-x) because it has the inner function 1-x. So, f'(0)=-f'(1-0)=-f'(1). No?
In order to solve the second problem, you'll need to know the following: A (nonempty) subset Y⊂Z is a subgroup of Z if it satisfies two conditions:
1) Y is closed with respect to the operation of Z
We know that the operation is addition, so in order for Y to be closed, then for each x,y∈Y then x+y=w, where w∈Y. EX: Say Y is the set of natural numbers under addition, then it is closed because the sum of any combination of natural numbers will always be a natural number. But, if Y is the set of natural numbers under subtraction, it is not closed because 3-8 results in a negative number.
2) for each y∈Y there exists a y^(-1)∈Y
This statement is referring to the existence of an additive inverse (because the operation is addition) for each element in Y. You'll also have to know the identity element, which for addition is zero because for each x∈Y, x+0=x. That is, the identity element leaves x unchanged. Since we know what an inverse does, we know that y+y^(-1)=0. EX: If Y is the set of natural numbers, then additive inverses do not exist because 3+(-3)=0. But, if Y is the set of integers, then we have additive inverses because this will include the negatives.
The first question is answered by Euclid's first proof. The answer is 120 degrees. The second question I can't answer. I haven't studied group theory yet so you'll have to fill me in. The last question is simple. If f(x)=f(1-x), then f'(x)=-f'(1-x) because it has the inner function 1-x. So, f'(0)=-f'(1-0)=-f'(1). No?
ReplyDeleteCorrect and correct.
ReplyDeleteIn order to solve the second problem, you'll need to know the following: A (nonempty) subset Y⊂Z is a subgroup of Z if it satisfies two conditions:
1) Y is closed with respect to the operation of Z
We know that the operation is addition, so in order for Y to be closed, then for each x,y∈Y then x+y=w, where w∈Y.
EX: Say Y is the set of natural numbers under addition, then it is closed because the sum of any combination of natural numbers will always be a natural number. But, if Y is the set of natural numbers under subtraction, it is not closed because 3-8 results in a negative number.
2) for each y∈Y there exists a y^(-1)∈Y
This statement is referring to the existence of an additive inverse (because the operation is addition) for each element in Y. You'll also have to know the identity element, which for addition is zero because for each x∈Y, x+0=x. That is, the identity element leaves x unchanged. Since we know what an inverse does, we know that y+y^(-1)=0.
EX: If Y is the set of natural numbers, then additive inverses do not exist because 3+(-3)=0. But, if Y is the set of integers, then we have additive inverses because this will include the negatives.