Question 1: Let L1, L2, ..., Ln be distinct lines in the Euclidean plane, and let A be the set of points formed by intersections of these lines. Characterize A using set notation and quantifiers.
Question 2: Express (as simply as you can) each of the subsequent sentences without the use of universal quantification:
(a) (?x)(?y)(?z)[P(x, y, z)]
(b) (?y) [(?x) P(x, y )? (?x ) Q(x, y )]
Question 3: Express (as simply as you can) each of the sentences in problem (2) without the use of existential quantification.
Question 4: A sequence of natural numbers ( a1, a2, ..., an ) is said to be a degree sequence if there exists an undirected graph on vertices { v1, v2, ..., vn } such that degree ( vi) = ai for each i = 1, 2, ..., n.
Part 1 Is (0, 1, 1, 1, 2, 2, 3, 4) a degree sequence? Prove your answer.
Part 2 Is (0, 1, 1, 1, 2, 3, 3, 4) a degree sequence? Prove your answer.
Part 3 FOR EXTRA FUN: try to devise an algorithm for determining whether a given sequence of numbers is a degree sequence.