By Irena Swanson

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And so on for the subsequent columns. Let the entry in the nth row and kth column be denoted nk . We read this as “n choose k”. These are loaded words, however, and we will eventually justify these words. Pascal’s triangle is defined so that for all n ≥ 1 and all k = 0, 1, . . , n − 1, n n + k+1 k n+1 . k+1 = What would it take to compute 100 ? It seems like we would need to write down rows 5 0 through 100 of Pascal’s triangle, or actually a little less, only slanted columns 0 through 5 of these 101 rows.

Consider the Cartesian product N0 × N0 . Elements are pairs of the form (a, b), with a, b ∈ N0 . If a, b, c, d ∈ N0 , we will write (a, b)R(c, d) if a + d = b + c. Thus R is a relation on N0 × N0 . (Certainly you are familiar with Z, in which case you may want to think of this relation simply saying that (a, b)R(c, d) if a − b = c − d. ) (1) R is reflexive: because for all (a, b) ∈ N0 × N0 , by commutativity of addition, a + b = b + a, so that by definition of R, (a, b)R(a, b). (2) R is symmetric: if (a, b)R(c, d), then by definition a + d = b + c, so that by commutativity of addition, d + a = c + b, and by symmetry of the = relation, c + b = d + a.

As a subset of R × R, ≤ consists of all points on or above the line y = x, and is represented (partially) as the shaded region in (2) Another relation on R: R = {(a, b) : a, b ∈ R, a2 < b + 1}. (3) Draw anything in R × R. That defines a relation on R (which most likely cannot be expressed with a formula). (4) Some relations on the set A of all people are cousin, parent, older than, has same birth date, and so on. 3: Relations, equivalence relations 61 {(2, a)}, {(2, b)}, {}. 3 Relation names such as “≤” or “cousin” are familiar, but most relations do not have such familiar or even good names; think of relation on A and B simply as a possibly contrived subset of A × B.