Suppose that T is a set of formulas and A and B are two formulas. (a) Show that if TE AA B, then IF A and TF B. (b) Is it true that that IFAV B, then IE A or IF B? If yes, then give a (meta)proof. If no, then give counterexample for appropriately chosen I', A, B.
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- In the WHILE language, prove that if <while b do y := y-x,s> ⇓ s' then there exists an integer k such that s(y) = s'(y) + k * s(x) Please use induction on derivations, with induction hypothesis.Prove: Let a, b, and c be integers. If (a - b) | c, then a | c. bk Suppose a, b, c are integers with (a b) | c. Then c = .). (² X . (a. b) for some integer k, so c = (alel X a, so a c.Given A = {1,2,3} and B={u,v}, determine. a. A X B b. B X B
- Let ƒ : Z → Z be some function over the integers. Select an appro- priate proof technique (direct, contradiction, induction, invariant or pigeonhole), and prove that if you have: a, f(a), ƒ(f(a)), ƒ(f(f(a))) and f(f(f(f(@)))), then at least 2 of those numbers are equivalent modulo 4.Let A={+,x,a,b}. Show that (a*V ba)+ b is regular over A.Suppose that the equation ax b .mod n/ is solvable (that is, d j b, whered D gcd.a; n/) and that x0 is any solution to this equation. Then, this equation has exactly d distinct solutions, modulo n, given by xi D x0 C i.n=d / fori D 0; 1; : : : ; d 1
- For (∃ x)(P(x,b)) Would an example of this being true if the domain was all the Avengers and x was green skin, then "b" being the Hulk would make this true. Am example of this being false would be: If the domain was all integers and x was positive, even integers and "b" was integers greater than zero.Prove that A, (Vx)(¬A v B) E (Vx)B, if x is not free in A.We have been working extensively with the predicate "eventually greater than" defined on pairs of functions f of g. Which of the following is equivalent to f(x) is not eventually greater than g(x)? (Select all that apply) Group of answer choices ¬((∃x0)(∀x)(x>x0→f(x)>g(x))) ((∃x0)(∀x)(x>x0→f(x)≤g(x))) (∀x)(∃x0)(x0>x→f(x0)≤g(x0)) (∀x)(∃x0)(x0>x→f(x0)≤g(x0))
- ; 3. Let T U V and S: V W. We have that SoT: U → W. Prove that (SoT)* = T* o S*. 4. Let v₁,.., Un EV be a basis of V. (a) Define the dual basis vi,.., v € V*. (b) Prove that formula: a = Σi-na (v₁) vi for every a € V*. i=1:n (c) Prove the reciprocal formula: v= Ei=1:n v (v) v₁ for every v € V.Prove valid. (x) ( ~ Ax > Dx) / (Ex) (Ax v Bx) (Ey) ( ~ (By v ~ Dy) > ~ ( ~ Dy v ~ Ay)) (x) ((Ax v Dx) > ( ~ Ax v ~ Dx))Prove: Let a, b, and c be integers. If Suppose a, b, c are integers with (a (a - b) | c, then a | c. b) | c. Then c = (a b) for some integer k, so c = ]a, · so a c.