Recent content by 4Fun

I'm currently reading the section on relations in Velleman's "How to prove it" and I have found a statement somewhere that I want to prove, but I'm not sure whether what I have come up with is reasonable and I also have some questions on the logic used in these type of proofs. The theorem is...

Ah yes, didn't think of that option. Thanks a lot for your help. In general if you try to prove equalities like this, what should you think about? Should you just consider all definitions that are valid and then just try some things out until you come up with the desired equality or what...

Homework Statement Let (G,\circ) be a group. Show that \foralla\inG (a^{-1})^{-1} = a The Attempt at a Solution I came up with the following. a^{-1} is the inverse of (a^{-1})^{-1}, therefore we have: a^{-1} \circ (a^{-1})^{-1} = e But for a^{-1} \circ (a^{-1})^{-1} to be equal to...

Ok thanks a lot. I did as you said and when solving for y I got a term with a11a22 - a12a21 in the denominator. So if a11a22 - a12a21 != 0, then the term is defined and there are unique solutions for y and x. If a11a22 - a12a21 = 0, the system would have no solutions, and I'm not sure about the...

Thanks for your quick reply. We haven't covered matrices and determinants yet. Is there another way, without using determinants to prove this?

Homework Statement Prove that the linear system of the form: a_{1,1}x + a_{1,2}y = b_{1}\\ a_{2,1}x + a_{2,2}y = b_{2} has exactly one solution in x and y if a_{1,1}a_{2,2} \neq a_{1,2}a_{2,1} The Attempt at a Solution I proved the contrapositive: Assume that the...

Homework Statement Prove that replacing one equation in a system of linear equations by a non-zero multiple of itself does not change the solution of the system. The Attempt at a Solution I'm still relatively new to proofs, so this is what I have come up with: Let S be a system of...

Thanks a lot for the quick answer. One more question though, what does it mean for an equation to be linear in x, y and z?

Homework Statement Let k \in \Re / {0}. Which of the three equations is linear in x, y, z? x + y - z = tan(k) kx - (1/k)y = 6 (3^k)x + y + z = 12 Well the question is basic, but I'm a little stumped because the questions asks for a single linear equation. By definition a linear...

Hello guys, this is my first post on this forum. I want to learn advanced/pure mathematics basically just because I find it really interesting and challenging and I have started to learn about proofs. I'm currently reading Velleman's book and I have reached the part in which you actually...