MATH 335 Assignment 2 Solutions

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5. Prove that there are 10 y’s in the system.

Proof: Axiom 2 and axiom 3 taken together show that there exists exactly one y for each pair of distinct x’s and no other y’s. How many pairs of distinct x’s are there? Exactly 10.

6. Prove that any two distinct y’s have at most one x on both.

Proof: Suppose that two y’s have at least two distinct x’s on both of them. Then by axiom 2 the two y’s are not distinct. This proves the result by contradiction.

7. Prove that not all x’s are on the same y.

Proof: Axiom 1 says that there are five x’s. Axiom 3 says that any y has exactly two x’s on it. Since 2 does not equal 5, not all the x’s are on the same y.

8. Prove that there exist exactly four y’s on each x.

Proof: Let A be a given x. Then let B, C, D, and E be the remaining four x’s. There are then four y’s on x determined by the pairs of points (A, B), (A, C), (A, D), and (A, E). Any other y on A would have to be on another x. This x could not be B, C, D, or E for that would violate Axiom 2. Thus since there are exactly 5 x’s there is no such other y.

9. Prove that for any x₁ and any y₁ not on that x₁ there exist exactly two other distinct y’s on x₁ that do not contain any of the x’s on y₁.

Proof: y₁ is on two x’s – say x₂ and x₃. There are two other x’s in the system – say x₄ and x₅. The pairs (x₁, x₄) and (x₁, x₅) determine a pair of distinct y’s. Any other y on x₁ has to also be on either x₂ or x₃. Thus there are exactly two y’s satisfying the theorem.