Solutions of two of the exercises that I gave in class.

Is the relation R reflexive? symmetric? transitive? (with justification). Find a graph such that the associated relation (obtained by folloiwng the same method as above) is an equivalence relation. What are the equivalence classes of this equvalence relation?

Composing the above bijective correspondences yields a bijective correspondence between the set of irreducible Pythagorean triples and the set ℚ∩(0,1). Use this to give an example of an irreducible Pythagorean triple that is not of the form (2n-1,2n

Two practice exams (one with solutions, one without)

1 ↔ ± n_{1} . a_{1} b_{1} c_{1} d_{1}....... | ( n_{i} ∈ ℕ∪{0}, | |

2 ↔ ± n_{2} . a_{2} b_{2} c_{2} d_{2}....... | a_{i}, b_{i}, c_{i} etc are | |

3 ↔
± n_{3} . a_{3} b_{3} c_{3} d_{3}....... | decimal digits ) |

such that

Infinite unions, infinite intersections of sets, and various notations for them.

((A∪B)-(C∪(A∩B)))∪(A∩B∩C) = (A-(B∪C))∪(B-((A∪C)-(A∩C))).

• Show with the help of Venn diagrams that for every four sets A, B, C, D, we always have

(A∩C)-(B∪D) ⊆ ((A∪D)∩(C∪B))-((C∩D)∪(A∩B)), but that this inclusion is typically not an equality.

Draw a Venn diagram for the sets A={1,2,6,7,8,9,10,11}, B={2,3,4,6,10,11,12,14}, C={4,5,6,7,8,11,12,13} and D={8,9,10,11,12,13,14,15}.