Consider the schema R = (A, B, C, D, E, G) and the set F of functional depen- dencies: AB → CD ADE → GDE B → GC G → DE Use the 3NF decomposition algorithm to generate a 3NF decomposition of R, and show your work. This means: a. A list of all candidate keys b. A canonical cover for F, along with an explanation of the steps you took to generate it

Q: Use logical equivalences to show that (p → q) → (notq → notp) is a tautology. Could you please show me this without just stating C2 rule (ϕ → ψ ≡ notψ → notϕ)   this is what we did in lecture but I am confused.  (p → q) → (notq → notp) (p -> q) -> (p->q) not(p->q) v (p->q) (p->q) v not(p->q) T

1) Find all candidate keys 2)Find a canonical cover of the FDs

Give conditions under which the following expressions are equivalent:A,Bγagg(C)(E1 ⋈ E2) and (Aγagg(C)(E1)) ⋈ E2where agg denotes any aggregation operation. How can the above conditionsbe relaxed if agg is one of min or max?

Examine the following argument: 1. ∀z (T(z) ∨ R(z))        Premises2. T(g) ∨ R(g)        Universal instantiation from (1)3. T(g)            Simplification from (2)4. ∀z T (z)            Universal generalization from (3)5. R(g)            Simplification from (2)6. ∀z R(z)            Universal generalization from (5)7. ∀z (T(z) ∨ ∀z R(z))    Conjunction from (4) and (6)The above argument shows that if ∀z(T(z)∨R(z)) is true then ∀z T (z) ∨ ∀z R(z) is true. Is this argument correct? Why or Why not? If it is incorrect, identify the error or errors.

Pick the option that is correct for A, B, C, D (A) Element defintion or Hypothesis  (B) Exsistential generaliztion 1, 2 or Exsistential instantaion 1, 2 or Universal generaliztion 1, 2 or  Universal  instantaion 1, 2 (C) Q(b) ^ R(b) or  Q(b) ∨ R(b) or Q(x) ^ R(x)  or  Q(x) ∨ R(x) (D) ∀Q(x) or ∀¬Q(x) or ∃Q(x) or ∃¬Q(x)

Dissect NP and P, then provide concrete instances to show how they vary in use.

Provide an interpretation that is true and one that is false: (∀ y)(P(y) → P(b))

please explain step by step

Use the Pumping Lemma to show that the the following language over {a,b,c}* is not CF  {wct | t ∈ {a,b}* ∧ w is a substring of t}