Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Problem Set 4

Module number: BEE3054

Problem 1. As noted in lecture, sometimes Vx e X : P(x) is written as Vx : P(x), with the set X understood. Please show that this alternate format can also be accomplished by re-writing the predicate. That is, given the sentence Vx e X : P(x), find an alternate predicate P'(X) such that we can write Vx : P'(X) without having to assume what the set X is, meaning that P'(x) satisfies

Vx e X : P(x) o Vx : P'(x)

Problem 2. Please repeat the exercise from problem 1 but for the existential quantifier. That is, find a P'(x) such that

3x e X : P(x) o 3x : P'(x)

Problem 3. Consider the following predicate logic sentence.

ExVyVzEtVs : P(x, y, z, t, s)

1. Please write down the negation of this sentence.

2. Please write down all sentences that are logically equivalent to this sentence.

Problem 4. Please prove that the discrete metric (d(x, y) = 0 if x = y, d(x, y) = 1 if x = y) satisfies the three axioms of a metric for any set X.

Problem 5. Prove that the British rail metric (d(x,y) = 0 if x = y, d(x,y) = ||x|| + ||y|| if x = y, where ||x|| is the Euclidean distance from the origin) is a metric for X = Rⁿ for arbitrary n.

Problem 6. Consider the set X = {0,1}ⁿ (the set of sequences of 0's and 1's of length n).

1. Provide an example of an element x e X with n = 4

2. Consider the function d(x,y) = {i e {1,2,...,n}|x《= y《}. Discuss this function and prove whether or not it is a metric.