MIT6_042JF10_rec16

MIT6_042JF10_rec16 - 6.042/18.062J Mathematics for Computer...

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6.042/18.062J Mathematics for Computer Science November 5, 2010 Tom Leighton and Marten van Dijk Problems for Recitation 16 1 Combinatorial Proof A combinatorial proof is an argument that establishes an algebraic fact by relying on counting principles. Many such proofs follow the same basic outline: 1. Define a set S . 2. Show that | S | = n by counting one way. 3. Show that | S | = m by counting another way. 4. Conclude that n = m . 2 Triangles Let T = { X 1 , . . . , X t } be a set whose elements X i are themselves sets such that each X i has size 3 and is ⊆ { 1 , 2 , . . . , n } . We call the elements of T “triangles”. Suppose that for all “edges” E ⊆ { 1 , 2 , . . . , n } with | E | = 2 there are exactly λ triangles X T with E X . For example, if we might have the triangles depicted in the following diagram, which has λ = 2, n = 4, and t = 4: 1 2 3 4 In this example, each edge appears in exactly two of the following triangles: { 1 , 2 , 3 } , { 1 , 2 , 4 } , { 1 , 3 , 4 } , { 2 , 3 , 4 }
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Recitation 16 2 Prove n ( n 1) λ = 3 t · 2 by counting the set C = { ( E, X ) : X T, E X, | E | = 2 } in two different ways.
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