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Problem 1 [8pt] Prove that the following two Hoare triples are valid. (Hint: in predicate logic...
prove that the arguments are valid using rules of inference and laws of predicate logic, (state the laws/rules used) Væ(P(x) + (Q(x) ^ S(x))) 3x(P(x) R(x)) - - .. Ex(R(x) ^ S(x)) - - - (0)H-TE. - – – – – – (24-TE ((x)S_(w))XA ((x)S ^ ()04)XA (2) 1 (x)d)XA
Suppose the domain of the following predicate logic propositions is {1, 2, 3}. Express the following statements without the use of quantifiers-only conjunctions and negations. a) b) Vx(( 3)P(x)) V P(x) Va, у(Р(2) —> (г. у))
Question 19 Prove the following statement: 1.1!+2 2!+...+d. d! = (d + 1)! – 1 when d > 0, d is an integer. B 1 y A - A IX E x x ili . TT. 12pt
in a laurent series valid (z-2) (1+2) 2+11 > 2
Simplify the following sentences in predicate logic so that all the negation symbols are directly in front of a predicate. (For example, Vx ((-0(x)) + (-E(x))) is simplified, because the negation symbols are direct in front of the predicates O and E. However, Væ -(P(2) V E(x)) is not simplified.) (i) -(3x (P(x) 1 (E(x) + S(x)))) (ii) -(Vx (E(x) V (P(x) +-(Sy G(x, y))))) Write a sentence in predicate logic (using the same predicates as above) which is true...
PROBLEM 2 Consider the family of circles P = {C, TER>0}, where Cr = {(x, y) R2 | x2 + y2 = p2} is a circle of radius r > 0. Prove that P is a partition of R2. State an equivalence relation induced by this partition. Hint: What is a property that is True for all points in a fixed circle?
Problem 2. Let f be a self-map on a set X. For x,y e X define x ~ y if and only if f"(x) = f(y) for some integers n, m > 0. Show that ~ is an equivalence relation.
9. Let x,y > 0 be real numbers and q, r E Q. Prove the following: (а) 29 > 0. 2"а" and (29)" (b) x7+r (с) г а — 1/29. 0, then x> y if and only if r4 > y (d) If q (e) For 1, r4 > x" if and only if q > r. For x < 1, x4 > x* if and only if q < r.
(2) Prove by induction that for all integers n > 2. Hint: 2n-1-2n2,
3. Suppose that X has pdf fx(x) = 3, x > 1 and Y has pdf 24» fy(y) = ¡2, x 〉 1. Suppose further that X and Y are inde- pendent. Calculate the P(X 〈 Y).