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  1. K

    What is the field with one element ?

    What is the "field with one element"? From the definition of a field, it follows that such a thing does not exist. However a Google search reveals that apparently there is, or at least mathematicians think there ought to be, something that goes by that name. What is it? Is it called a "field"...
  2. K

    Induction question

    Prove by induction that 2^{2^n}+1 always ends in 7 for all n > 1 (true for n = 2). I couldn't figure out anything to do with the last digit being 7, so I looked the case that 2^{2^n} ends in 6 for all n > 1, which is also true for n = 2. Suppose it's true for n = k: 2^{2^{k+1}}=2^{2^k\cdot...
  3. K

    Pot. energy

    If h, the height of a body from the ground, approaches infinity, then the body's potential energy approaches 0? I'm assuming a non-constant value for g. E_p = mgh. In this equation I get 0 times infinity, which is mathematically indeterminate, but I'm guessing that the physical interpretation...
  4. K

    Quick question

    The antisymmetric relations on a set {a,b} are those, which do not contain both of the pairs (a,b) and (b,a) because that would imply a = b, however a can't equal b since they are elements of a set? PS: In our course we allow only one copy of an element in a set, so {a,b} is a set only if a...
  5. K

    A proof of relations

    I answered this wrong on a test, but now I've come up with a different solution. Problem: Prove that a relation xRy\Leftrightarrow x-y\in\mathbb{Z} defined on \mathbb{R} is an equivalence relation. Solution: 1.) Reflexivity: xRx,\forall x\in\mathbb{R} For every x we have x-x=0 which is an...
  6. K

    Function proof

    I had this question on a test today. Prove that if a function f:X-->Y is injective, then f(X\setminus A) \subset Y\setminus f(A), \forall A \subset X. This is how I did it: If x_1 is in A, then y_1=f(x_1) is in f(A). Because the function is injective, we can pick (cut Y into pieces) f(A) and...
  7. K

    Limit of absolute value

    Is it correct that |\sin x|=-\sin x as x\rightarrow 0-? It's clear for a function like y=x, but I ask the question since the sine oscillates from positive to negative values, so for different x's the abs. value is either pos. or neg. Or do I only need to consider values of x that are between...
  8. K

    Universal set question

    Let's say I have to prove something about two sets and I want to make use of the notion of the universal set. Is it OK then to define the universal set as the union of these two sets? - Kamataat
  9. K

    Sup&inf question

    Quote from my textbook: "For any two sets of real numbers, X=\{x\} and Y=\{y\}, the following hold: \sup(x-y)=\sup x-\inf y (1), \inf(x-y)=\inf x-\sup y (2), where x\in X and y\in Y." It is also said, that if some set A consists of some elements a, we may write it as A=\{a\}, and...
  10. K

    Mass of water molecule

    Is there any way to find the mass of a water molecule when only Avogadro's number is given? Thanks! - Kamataat
  11. K

    Eraser sliding down slope - Friction

    An eraser slides down a slope of height 1m that makes an angle of 35 degrees with the ground. It takes the eraser 1,8s to do this. Find the coefficient of friction. I have no clue how to do this. I've not been able to solve any of the equations I've tried, because there is always the mass of...
  12. K

    What force does the dynamometer display?

    Hi! Really simple question: Two people are pulling on the ends of a dynamometer, each with a force of 25N. What force does the dynamometer display? Textbook answer: 25N I'm having a difficult time understanding this conceptually. Could anybody explain why the answer is 25N? Thanks...
  13. K

    Trig proof question

    Let \alpha,\beta\in]0;\pi/2[. Prove that \sin(\alpha+\beta)<\sin\alpha+\sin\beta. My intuition says it's true, as can be seen from \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta, because 0<\cos\alpha,\cos\beta<1, but I haven't been able to prove it. Thanks in advance!!! -...
  14. K

    Double intergrals

    To calculate the volume of a cylinder that has as its bottom (or top) end the area D in the xy-plane, we divide D into n smaller areas D_i (i=1;...;n). The function f(x,y) is defined at every point P(x,y) of D, in short f(P). So, to find the volume V_i that is above an area D_i, we use the...
  15. K

    Really simple question

    I didn't want to post this in the logic subforum, because it's really basic... I know that when proving mathematical statements, one can't assume what he/she is trying to prove. But can one assume its converse? - Kamataat
  16. K

    Please check this

    Verify A\cap(B-C)=(A\cap B)-(A\cap C). This is how I did it: From x\in[A\cap(B-C)] (lhs) we have that 1.) x\in A 2.) x\in B 3.) x\notin C From #1 and #2 we have that 4.) x\in(A\cap B) Now, since x\in(A\cap C) means that x\in A and x\in C, but we have x\notin C, it follows that 5.)...
  17. K

    Trig proof

    I have to prove that a/\sin\alpha =b/\sin\beta. A triangle has sides "a", "b", "c" and angles \alpha and \beta (opposite of the sides "a" and "b" respectively). This is what I did: Draw a line "h" as the height of the triangle on the side "c". \sin\alpha = h/b. Multiplying by "b" gives...
  18. K

    Are these OK to read?

    "Gödel, Escher, Bach: An Eternal Golden Braid" by Douglas. R. Hofstadter and "The Physics of Consciousness" by Evan Harris Walker. The second one sounds dubious, but I'd like a few opinions from people who know more about science than me. The reason why I don't want to go by Amazon reviews is...
  19. K

    Time of boat completing round trip

    A boat made a round trip of 3km on a river in 2h. The speed of the river's flow was 1m/s. In what time will the boat complete the same round trip, if the speed of the flow is 2 times smaller? I suck at these kinds of questions, so would anybody please help me out here? - Kamataat
  20. K

    Relative motion

    Hi, I have this excercise that I've been trying to do for a couple of days now. Can anybody help me? Here's the question: A boat is moving against the direction of flow of a river and meets a piece of wood that is moving along with the river. The boat continues against the flow of the river...
  21. K

    Inequality Proof help

    This isn't homework or anything. I'm just interested in learning to prove things in mathematics, so I took a piece of paper and did the following: For every a < b it is true that a < (a+b)/2 < b (1). First I separated the inequality: a < (a+b)/2 and (a+b)/2 < b. Then I did this: a <...
  22. K

    Concerning motion

    Hi, I was given the following question: Q: A body starts moving with an acceleration a=const. After a time t it has covered the distance \Delta x[/itex]. What is its speed at time t? Answer: \frac{2\Delta x}{t} Can somebody say how to arrive at this answer (both mathematically and...
  23. K

    Is this correct?

    Find the interval in which the function y=x+\sin x\cos x is increasing. So, first I differentiated to get y'=1+\cos 2x. Then I set y' equal to zero: 1+\cos 2x=0 \cos 2x=-1 2x=\pm \arccos m+2n\pi, where n\in\mathbb{Z} 2x=\pm \arccos(-1)+2n\pi 2x=\pm\pi+2n\pi x=\pm\frac{\pi}{2}+n\pi So...
  24. K

    Increasing/decreasing f(x)

    Let's take y=x^2 as an example. This function decreases if -\infty < x < 0 and increases if 0 > x > \infty. But what about x=0? Shouldn't it be included in one of the two ranges of x? - Kamataat
  25. K

    Logarithm question

    Is it true that that ln(-x) is defined for x \in R such that x < 0? - Kamataat
  26. K

    Simple set question

    I need to prove that -(-A)=A. I guess it's the same as S-(S-A)=A, where S is the space. So is it true, that if x \in S-(S-A) then x \notin S-A? - Kamataat
  27. K

    Something to be a subset of something?

    The book I'm reading says that from p \in A and A \in M it does not follow that p \in M, if M is a family of sets and p is an element of A. However, then further down on the same page it says that for any sets A, B, C it is true that if A \subseteq B and B \subseteq C, then A \subseteq C...
  28. K

    Maxima-minima problems

    Hi! For example y=-x^3-3x=0 gives y'=-3x^2-3 and setting y'=0 we get i and -i as the solutions. What does this say about the existence of the max and min points for the function y? - Kamataat
  29. K

    Astronomers observe some galaxy

    Hi! Let's say that astronomers observe some galaxy or something several million lightyears away and then come to conclusions about that object. But since the light from there has been on the way for millions or even billions of years, then is it correct, that they are actually seeing the...
  30. K

    Anthropic principle?

    Hi! I don't really know if this is the right place to post this, but here's the question: It says that the universe is tuned so that we can exist. So if it were any different, we wouldn't be here to observe it. But why? I mean, there could still be life, maybe even more intelligent than us...
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