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MAT 141
Homework 8 – solutions
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- Logic - Assume that the following statements are true. David, Elizabeth and Huy are on the soccer team. Kim, Mimi and Parker are on the golf team. Stewart and Samir are on the robotics team. The people in the group are David, Elizabeth, Huy, Kim, Mimi, Parker, Stewart and Samir.
Decide whether each of the following are true, false or not determined by the information given:
a: Everybody in the group is on a team. true b: Nobody in the group plays chess. not determined by the information given c: If somebody in the group plays soccer, then somebody else plays golf. not determined by the information given
Negate statements a, b, c.
Negation of a: Somebody in the group is not on a team. Negation of b: Somebody in the group plays chess. Negation of c: Somebody in the group plays soccer and nobody else plays golf.
- Proposition 4.7: Hilbert’s Parallel Postulate holds if and only if every line intersecting one of two parallel lines also intersects the other.
Proof: We prove two implications. (“⇒”) Suppose that Hilbert’s Parallel Postulate holds, i.e., suppose that, given any line l and a point P not on l, there is at most one line parallel to l that is incident to P. Let l, l′^ be a parallel lines and and let m be a line incident to l in a point P. Suppose, to derive a contradiction, that m is disjoint from l′. Then both l and m are parallels to l′^ incident to P. This contradicts Hilbert’s Parallel Postulate. Thus m intersects l′. (“⇐”) Suppose that every line intersecting one of two parallel lines also intersects the other. Let n be a line and let Q be a point not on n. Suppose, to derive a contradiction, that there is more than one line parallel to n and incident to Q. Let n′, m′^ be two such lines. Then m′^ intersects n′^ in the point Q and n′^ is parallel to n. Hence by hypothesis, m′^ also intersects the line n. Thus m′^ is not parallel to n, a contradiction. Hence Hilbert’s Parallel Postulate holds. ∎
n through Q. Then t′^ is a transversal for n, m and also for n, m′. By hypothesis, alternate interior angles are congruent in both cases. By Congruence Axiom C-4 there is a unique ray ema- nating from Q on a given side of t′^ congruent to an interior angle formed by t′^ and n. Hence m and m′^ must coincide. Thus Hilbert’s Parallel Postulate holds. ∎.
Proposition 4.9: Hilbert’s Parallel Postulate holds if and only if whenever a transversal t meets a pair of parallel lines l, l′, if t ⊥ l, then t ⊥ l′.
Proof: We prove two implications. (“⇒”) Suppose that Hilbert’s Parallel Postulate holds, i.e., suppose that, given any line l and a point P not on l, there is at most one line parallel to l that is incident to P. Let l, l′^ be parallel lines and let t be a transversal to l and l′. Furthermore, suppose that t ⊥ l. By the converse to AIA, alternate interior angles are congruent. Hence, by the definition of perpendicular and right an- gle, all four interior angles for t, l, l′^ are right angles. Thus t ⊥ l′. (“⇐”) Suppose that whenever a transversal t meets a pair of parallel lines l, l′, if t ⊥ l, then t ⊥ l′. Let n be a line and Q a point not on n. Suppose that m, m′^ are parallels to n incident to Q.
By Proposition 3.16, there is a line t perpendicular to n and incident to Q. By hypothesis, t ⊥ m and t ⊥ m′. Thus m and m′^ must coincide. Therefore Hilbert’s Parallel Postulate holds. ∎.
Proposition 4.10: Hilbert’s Parallel Postulate holds if and only if for every pair of parallel lines k, l and lines m ⊥ k, n ⊥ l, either m = n or m is parallel to n.
Proof: We prove two implications. (“⇒”) Suppose that Hilbert’s Parallel Postulate holds, i.e., suppose that, given any line l and a point P not on l, there is at most one line parallel to l that is incident to P. Let k, l be parallel lines and m, n lines such that m ⊥ k and n ⊥ l. Case 1: m = n. In this case the conclusion follows. Case 2: m ≠ n. In this case, by Proposition 4.9, m ⊥ l. In particular, l is a transversal for the pair of lines m, n and has congruent alternate interior angles. By AIA, m, n are parallel. (“⇐”) Suppose that for every pair of parallel lines k, l and lines m ⊥ k, n ⊥ l, either m = n or m is parallel to n. Let l′^ be a line and P ′^ a point not on l′. Suppose that m′, m” are parallels to l′^ incident to P ′. By Proposition 3.16, there is a line t ⊥ l′^ incident to P ′. Denote the point at which t and l′^ meet by Q. By Proposition 3.16, there is a line t′^ ⊥ m′^ through Q.
