Coding Theory and Design Theory. Coding Theory by Ray Chaudhuri

By Ray Chaudhuri

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179 Example Find the number of surjections from A = {a, b, c, d} to B = {1, 2, 3}. Solution: The trick here is that we know how to count the number of functions from one finite set to the other (Theorem 177). What we do is over count the number of functions, and then sieve out ¡ those which are not surjective by means of Inclusion-Exclusion. By Theorem ¡ 177, there are 34 = 81 functions from A to B. There are 31 24 = 48 functions from A to B that miss one element from B. There are 32 14 = 3 ¡ functions from A to B that miss two elements from B.

Then • the number of functions from A to B is mn . • if n ≤ m, the number of injective functions from A to B is m(m − 1)(m − 2) · · · (m − n + 1). If n > m there are no injective functions from A to B. If a function from A to B is injective then we must have n ≤ m in view of Theorem 174. , and so we have m(m − 1) · · · (m − n + 1) injective functions. ❑ 178 Example Let A = {a, b, c} and B = {1, 2, 3, 4}. Then according to Theorem 177, there are 43 = 64 functions from A to B and of these, 4 · 3 · 2 = 24 are injective.

Dividing by 73 this last equality we obtain 1+ proper fraction = a3 + proper fraction, and so a3 = 1. Continuing in this way we deduce that 5213 = 211257 . The method of successive divisions used in the preceding problem can be conveniently displayed as 7 7 7 7 7 5212 744 106 15 2 5 2 1 1 2 The central column contains the successive quotients and the rightmost column contains the corresponding remainders. Reading from the last remainder up, we recover 5213 = 211257 . 201 Example Write 5627 in base-five.

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