Question 3# (Cardinality) Define a relation on N² = Nx N by: { a+by+z or a+by+z and a

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Question 3 Cardinality Define A Relation On N Nx N By A By Z Or A By Z And A Y For Example 3 2 1 5 Becau 1 (75.48 KiB) Viewed 28 times

Question 3# (Cardinality) Define a relation on N² = Nx N by: { a+by+z or a+by+z and a <y. For example (3, 2) (1,5) because 5 < 6 and (1,5) (2, 4) because 1 < 2. (a) Justify each of the following properties of ': (i) If (a, b) and (y, z) are any two distinct pairs in N2 then either (a, b) (y, z) or (y, z) (a, b) but not both. (ii) If (a, b), (p, q) and (y, z) are any three pairs in N2 with (a,b) (p, q) and (p, q) (y, z) then (a, b) (y, z). (This property is known as 'transitivity'.) (b) Properties (i) and (ii) above mean that is a full order. According to this <-order, the 'least', or 'first' pair in N² is p₁ = (1, 1) and the next is p2 = (1, 2). Write out the next eighteen pairs p3,...P20 in this sequence. P1 = (1,1) P2 = (1,2) P3 = P4 = P5 = P6= P7 = TU P8 = Pg= P10 = P11 = P12 = P13 = P14 P15 P16 = P17 = P18 = P19 = P20 = (c) Explain why this 'ordering' of N2 by shows that N2 is countable. (i.e. N² has the same cardinality as N). (d) Is N³ countable? Justify your answer. (a, b) (y, z) →