Recent content by Calculuser

I think there is a typo above. You meant ##\delta## instead of ##c##, I guess. What is ##I##, then? Is it the domain that function ##f(x)## is defined on, or is it a set that is tacitly assumed to be defined within ##(0,\ \delta)## like ##0 < x < \delta##? Did you get my point? If the latter is...

We discussed above why we cannot use alternative definition(s) of the formal limit definition. What I mean by, is if we had extra ##x < \delta## condition in the original definition, $$\lim_{x \to c}f(x) = L \iff [\forall \epsilon > 0\ \exists \delta > 0\ \forall x \in I\ (x < \delta \land (0 <...

I think my confusion results from the concept of "domain of discourse", was not specified by me above in the formal definition. What I have eventually come to realize, yet I am still not sure of it, that if we define ##x \in \mathbb{R}## and ##\epsilon,\ \delta \in \mathbb{R}^{+}##, what you all...

That explanation is quite vague to me. If it says there is such a delta, so we can pick one, say ##\delta = 1##. Why not? What prevents us from such a selection? What if we used that ##\epsilon =100## value in the original definition, would that say it is discontinuous if it would? If so, how...

Correct me if I am wrong in my thinking: ##D[1]## fails for a bell curve, or in general non-bijective functions, in the application of ##\implies## connective since "if ##|f(x) - L|< \epsilon##, then ##0 < |x - c| < \delta##". That is, we first take ##|f(x) - L|< \epsilon## interval along the...

Formal definition (epsilon-delta definition) of limit is symbolically as follows: $$\lim_{x \to c}f(x) = L \iff [\forall \epsilon > 0,\ \exists \delta > 0,\ \forall x \in I,\ (0 < |x - c| < \delta \implies |f(x) - L| < \epsilon)]$$ Now I want to create alternative definitions out of this by...

What I mean by that is a derivation that contains vector and linear algebra properties used as in the derivation of similar problem, "family of lines passing through one point", as shown below even though irrelevant, but I thought useful to share: Let ##d_1:a_1x+b_1y+c_1=0## and...

What we want to infer from statements some other statement by some inference rule is what happens in proofs as I try to detail as much as possible through my basic knowledge on propositional calculus. If every element of ##U_1+U_2+...+U_m## can be uniquely written as a sum of...

In "Sheldon Axler's Linear Algebra Done Right, 3rd edition", on page 21 "internal direct sum", or direct sum as the author uses, is defined as such: Following that there is a statement, titled "Condition for a direct sum" on page 23, that specifies the condition for a sum of subspaces to be...

After re-reading the question in response to your replies above, I too noted its vagueness as some of you pointed out. The game is played as follows: six numbers, e.g. 1, 6, 18, 19, 8, 5, out of 1, 2, . . . 20 are first recorded, say on a piece of paper; and then then one player goes on to...

Question: "In a lottery game each player tries to guess right 6 numbers designated in advance by choosing randomly from among numbers from 1 to 20. Given that one player guessed right 5 numbers out of 6 that he/she picked, what is the probability of guessing right the 6 numbers?" The problem...

As I was flipping through pages of my analytic geometry book from high school, in circle section I stumbled across the formula of "family of circles intersecting at two points" with two circles (##x^2 + y^2 + D_1 x + E_1 y + F_1 = 0## , ##x^2 + y^2 + D_2 x + E_2 y + F_2 = 0##) known to intersect...

I agree on that part! However, I do not get why it was extra(?) stated that \mathcal{B}_i can also be an axiom in the original text of the author as said "...either \mathcal{B}_i is an axiom..." in addition to the things you said. Wouldn't it be sufficient to say that \mathcal{B}_i is either a...

What I mean by that is a set of anything related to the real of propositions, e.g. atomic formulas, statement forms, etc. from which we deduce a consequence with some propositions within. Because in the screenshot I posted above it does NOT say that ALL...

Did you mean the entire text or just \Gamma=\{\mathcal{B}_1,\mathcal{B}_2,...\mathcal{B}_k\}? Could you clarify a bit? Then \Gamma is like a universal set of any statements that are either true or false. In light of what I understood in the comment on your explanation of \Gamma above, that...