Assumption $d>2$ on Proposition 2.12 from Knapp’s Elliptic Curves

I’m going through Knapp’s book on elliptic curves and I got stuck in a minor detail.

This is a part of the proof of Proposition 2.12:

I could understand everything except for this little detail: Where are we making use of the assumption $ d>2$ ?

I will post some pictures about the references that the proof makes use of, in order for you to understand the whole argument.

Proposition 2.7 and identity (2.12):

Lemma 2.11:

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Prove or disprove the following proposition

$ L_1^*∪L_2^*⊆(L_1∪L_2)^*$

I actually disproved the opposite proposition $ [(L_1∪L_2)^*⊆L_1^*∪L_2^*]$ and my intuition tells me that this is actually true… I tried to show that the combinations of words in the right group $ [(L_1∪L_2)^*]$ obviously contain all the words from the left group $ [L_1^*∪L_2^*]$ and it’s also very plain to see, but I have no idea how to formalize it.

Any idea how to formally prove it? Thanks.

Discrete math – negate proposition using the quantifier negation

I’m asked to negate the following proposition using the quantifier negation rules. No negation operations are to appear before any of the quantifiers in the expression that is created. The issue is I’m not quite understanding what this means. All I’m given in my notes relating to negation quantifiers are the following formulas and their proofs:



I’m not quite sure how to take this information and apply it to a proposition or really, I don’t quite understand what my end goal of this question is supposed to be.

This is the proposition I’m given to work with. I’d appreciate if someone taught me step by step how to solve these types of questions. You can make up your own proposition if you’d like, but I’m really confused and would appreciate some sort of example. The results I found online seemed really complex and confusing.

∃𝑥 (𝐷(𝑥) → (𝐶(𝑥) ∨ F(x)))

Proving Proposition with Predicate Logic

If we have the proposition

\begin{align} &\text{Bob is a Babylonian}\ &\text{Bob is a Human}\ &\text{Therefore, some Humans are Babylonians} \end{align}

which translates to

\begin{align} &B(p)\ &H(p)\ &\text{Therefore, } \exists x (B(x) \land H(x)) \end{align}

how would one prove such a proposition? Would it look something like


\begin{align} &\exists x B(x) … (\text{premise }a)\ &\exists x H(x) … (\text{premise }b) \ &\exists x (B(x) \land H(x)) … \text{from premise a) and premise b) and since} A \land B \vDash A \end{align}

EGA I (Springer), Proposition

I do not understand one argument in the proof of Proposition in the new version by Springer of EGA I. When proving that the functor $ F$ is representable by $ (X, \xi)$ , where we obtained $ X$ by gluing the objects $ X_i$ representing the subfunctors $ F_i$ , Grothendieck shows for an arbitrary $ S$ -ringed space $ T$ , that there is a bijection of the form $ Hom_S(T,X) \to F(T), g \mapsto F(g)(\xi)$ . For the proof of the injectivity, he needs the argument that the fibre product $ F_i \times_F h_X$ , where $ h_X \to F$ is the natural transformation corresponding to $ \xi$ , is representable by $ (Z_i,(\xi_i’, \rho_i’))$ , where $ Z_i$ is isomorphic to $ X_i$ , $ \rho_i’:Z_i \to X$ is the canonical injection and $ \xi_i’ = F(\rho_i’)(\xi)$ . For this fact, he argues with condition (i) in the Theorem, i.e. that $ F_i \to F$ is representable by an open immersion. I cannot see how the representability of the fibre product by this tuple follows from this. Thank you for your help!

Proposition A.2.6.15 of Higher Topos Theory

I am reading Higher Topos Theory to learn the properties of $ \infty$ -categories. I found that Proposition A.2.6.15 is used in Proposition to give a construction of the covariant model structure. But Proposition A.2.6.15 uses some features that had not been introduced until Chapter 5, making its proof inaccessible before reading Chapter 5.

Is there any proof that avoids using those structures that are introduced later in this book?