shah_nosrat

There's a mistake in your equality; it's supposed to be [latex]2xy^2 - y^2 + y = 2yx^2 - x^2 + x [/latex], factoring both terms on each side of the equality we have: [latex] y(2xy -y + 1) = x(2xy -x + 1) [/latex], which we can argue for x to equal to y we should have [latex]2xy -y + 1 = 2xy - x + 1[/latex], which reduces to [latex]x=y[/latex] as required, hence, your function is injective.

From what I studied in discrete mathematics, relations have matrix representations, and besides the operations we are used in linear algebra, there are also boolean operations that can be performed on matrices.

I don't understand your vector representations. But from the definition of linear independence, it says the following, that the linear combination of the vectors; k_{1}v_{1} + k_{2}v_{2} + k_{3}w = 0, if k_{1}=k_{2}=k_{3}=0 solving this equation will give your solution.

Just to answer your last statement. Abstract Algebra deals with the study of mathematical structures called groups. To give an example between LA (Linear Algebra) and AL (Abstract Algebra) on their similarity (not in property, but on the approach, since it's algebra). In LA we have basis sets that spans a particular vector space, and how an entire vector space can be constructed by the basis set. Similarly in AL we have cyclic sets that generates an entire group! So their approach is similar, but as John puts it: that's why is called Linear Algebra. Besides, LA has immense applications; handwriting analysis, solving linear first-order differential equations, search engines use the mathematics from LA, Differential Geometry: representation of the coefficients of the first-fundamental form is in matrix form. Hope this clarifies things.

I haven't read the above book. But I do know that any computer scientist needs to have knowledge of Discrete Mathematics: As this will teach you naive set theory, logic, counting principles, Relations, Digraphs, Graph theory, Languages and finite - state machines and much more. Then you could complement it with the above mentioned book. When dealing with Definitions, axioms, theorems, and proofs. It's always a good idea to understand what a particular definition, axiom or theorem is saying and then going on to reading the associated proof to get a complete picture of whats going on. Memorizing is never a good idea.

Edward Witten Michio Kaku Andrew Wiles Stephen Hawking Roger Penrose Yes some of the above are Theoretical Physicists, but as ajb put it, they did spur modern mathematics.

I don't know if considering robots as its own species would be a good idea, not to mention it being an intellectual being/entity. We would then have to consider their robotic rights as well, and would give rise and debate to ethical considerations of how to deal with these new robotic beings or their species as a whole. But I do know that Japanese scientists creating realistic humanoid robots to assist us in our daily chores or life for that matter, they consider it as being the next evolution of the so-called "Personal Computer". Now to consider something as alive, they would have to satisfy certain prerequisites (Which I'm not really sure of) but I'm sure it exists.

Hi, try some books by Stephen Hawking: George's Secret Key to The Universe, and other science books for Kids. I don't know if they improve critical thinking, but it sure will encourage imagination and creative thinking in kids, which is also important in any field of science.

Hi, I came across this theorem and decided to prove it, as follows: Theorem: A set [math]A \subseteq R[/math] is bounded if and only if it is bounded from above and below. I would like the prove the converse of the above statement; If a set [math]A \subseteq R[/math]is bounded from above and below, then it is bounded. Let [math]M = |M_{1}| + |M_{2}|[/math]and using this preliminary result I proved earlier [math]-|a| \leq a \leq |a|[/math]. Now, [math]\forall a \in A[/math] we have [math] a \leq M_{1}[/math] ---> definition of bounded from above. and [math] M_{2} \leq a[/math] ---> definition of bounded from below. Using the result: [math]-|a| \leq a \leq |a|[/math]. Since [math]M = |M_{1}| + |M_{2}|[/math] is the sum of absolute value of [math]|M_{1}|[/math] and [math]|M_{2}|[/math], it is a big number. (trying to convince myself). Also, [math]-M = -(|M_{1}| + |M_{2}|)[/math], This is on the opposite of the spectrum. Now, [math]-M \leq -|M_{2}| \leq M_{2} \leq -|a| \leq a \leq |a| \leq M_{1} \leq |M_{1}| \leq M [/math]. [math]M = |M_{1}| + |M_{2}| \geq |M_{1} + M_{2}| \geq 0 [/math] ---> Which shows that M is positive by using the triangle inequality. Hence, [math]-M \leq a \leq M[/math]. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I'm excited for this proof, hopefully it's correct. Your help is once again appreciated.

Thank you Dr.Rocket. You are giving me confidence in art of proofing. Thank you Dr.Rocket. You are giving me confidence in art of proofing.

Okay, taking your advice. Forget about my previous attempt at the proof. Using Linear (total) ordering. If we take [math]\mathbf{U} \subseteq W[/math] ---> I need to invoke existence for it to make sense (To me anyway) suppose [math]a, b \in \mathbf{U}[/math] with the property of being the least elements for all elements in [math]\mathbf{U}[/math]. Now, because of the antisymmetry property of the linear ordering then, as follows: if aRb and bRa then a = b ---> Does this conclude the least element in the subset if it exist is unique? Your help is once again appreciated

Seattle's Calling - by Burn The Charts

Hi, This is the question that needs a proofs, as follows: Show that the smallest element of a nonempty subset of [math]\mathbf{W}[/math] is unique. My attempt at the proof, as follows: Let [math]\mathbf{U} \subseteq \mathbf{W}[/math], by the well ordering principle (WOP) we have that [math]a \in \mathbf{U}[/math] such that [math]a \leq x [/math] [math]\forall x \in \mathbf{U}[/math]. Now suppose [math]b \in \mathbf{U}[/math] such that [math]b \leq x [/math] [math]\forall x \in \mathbf{U}[/math]. Since [math]0 \leq x - a[/math] and [math]0 \leq x - b[/math] by definition. Now, [math]0 \leq x + x - (a + b)[/math] [math]a+ b \leq x + x = 1\cdot x + 1\cdot x = (1+ 1)\cdot x = 2x[/math] [math]a+ b \leq 2x[/math] [math]\frac{a + b}{2} \leq x[/math] . The only way that this inequality will hold [math] \forall x \in \mathbf{U}[/math] is when [math]a = b[/math] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Is the above reasoning and proof correct?

First of all keep the doubts and negative thinking aside. I believe that anyone is capable of great things as long as they're interested. If you are interested in Higher Mathematics then that's the first step to learning Higher Mathematics. Let me tell you this, that Higher Mathematics trains you to think analytically and critically about any problem presented to you. And remember to always have fun

I keep my books, because I can always use them for reference later on.

I appreciate your feedback, and I agree it starts with the consumers. I think the best way to do this is by proper education, and make the general public aware of the damage waste is doing to our planet. We really need to be aware of this fact, because it is our environment that supports us, not the other way around.

Hi, I am currently completing my Undergraduate degree in BSc(Mathematics), but I'm glad to help. My motivation was simple, I always was intrigued by the Mathematics. To pursue what they are most interested in the field of science, and to always keep an open mind. Using the techniques and methods of differential equations to develop a mathematical model, and it's qualitative analysis, for example Lokta-Volterra model. Hmmmm, don't have one. But if I had to choose it would K (potassium). I follow achievements in Mathematics, and thus, The Mathematician Andrew Wiles proved Fermat's Last Theorem; which was first conjectured by Pierre de Fermat in 1637. To always believe in your capabilities, and never take no for an answer. Hope this helps

You can ask the question under the Mathematics section; I presume your question deals with a problem in Applied Mathematics, post your question under that heading in the Mathematics section. Do this by starting a topic!

I wouldn't know about the courses at Oxford University, but I do know that Physics and Philosophy are a good combination for a degree. Physics courses are pretty standard around the world with Universities offering the following at 3rd year: Quantum Mechanics Statistical and Thermal Physics Nuclear and Atomic Physics Solid State Physics Computational Physics and the required laboratory sessions As to regards to Philosophy you will be introduced to subjects such as: Critical Reasoning and Argumentation The Philosophy of Science ....etc. The above mentioned courses in Philosophy is important for the following reasons, respectively: Also courses in Theoretical and Applied Ethics is useful. The above quotes are course descriptions offered by the University of South Africa. Regards.

The following are a required, as follows: Biology (Extended) Chemistry (Extended) Physics (Extended) Mathematics (Extended) Additional Mathematics Also try completing A-Levels in the following subject, as follows: Mathematics Biology Chemistry A-Levels gives you strong grounding and makes the transition to first year University smooth. Hope this helps and Best of Luck in you endeavors

A good book would be Fundamentals of Physics by Jearl Walker, and will accompany you up to first year Physics at University. The book comes in a bundle which covers: Mechanics, Electromagnetism, Heat and Modern Physics. The book requires working knowledge of Differential and Integral Calculus, which can be acquired using the book Calculus Concepts and Contexts by James Stewart, which will also accompany up to first year University. I wouldn't know much about aerodynamics books, but if you intend to learn the more advanced aerodynamics theory, you would need a strong background in Mathematics in topics such as; Multivariate Calculus, Linear Algebra, Differential Equations. My suggestion would be to first learn the qualitative knowledge of aerodynamics, more like what pilots are introduced to while training for their PPL (Private Pilots License), for example what an airfoil is, its camber, leading and trailing edge, flaps spoilers. Also how the airfoil produces lift relative to moving air, the Venturi effect etc. It will benefit you as it will give a strong conceptual understanding of aerodynamics. Hope this helps and have fun

I know about the Honors/MSc and PhD Programs at the University of Cape Town, South Africa. There is a group within the Department of Mathematics and Applied Mathematics called MARAM (Marine Resource Assessment and Management Group) They also offer courses in Biological, Ecology and Environmental Modeling. http://www.mth.uct.ac.za/maram/ IELTS is usually an overall score of 7.0

Let me be frank, I'm taking a course in Programming this semester, and let me tell you it is boring! As you mentioned, that getting around solving problems isn't the issue, it's just that it's boring. Also if you feel as though you're not enjoying it, switch majors and do something you love, and that's what you'll excel at. If you like Mathematics, then pursue it. Mathematics is my major and I'm loving every minute of it . Hope this helps, and all the best. Regards.

Thanks!