Advanced methods in applied mathematics; lecture course by Richard Courant, Charles de Prima, John R. Knudsen

By Richard Courant, Charles de Prima, John R. Knudsen

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Thus IA n BI = 4. 1 by noting that 14+9-4=19. 1. 6 Case i. Suppose that x is a member of only one set, say A. A n BI, since A nBs B. Thus x contributes a count 0' one to iAI + iBI - IA n BI. If x is in B. but not in A, a similar argument suffices. Case ii. { and B. lee in 1,,4 n BI. BI- iA II BI, just as we wanted. 3. How many ,even-digit telephone numbers are there that begin 584- and contain at least one 0 and at least one I'? \ND COUNTING set of all such numbers that contain a 0 and B be the set of all numbers that contain a 1.

With domain and target the positive integers. that is onto and is not the identity map. 48 1:9 FUNCTIO,,"S 4. If two functions are not equal, they are called different. Suppose that A = [al,a 2, ... ,an]' How many different functions are there with domain A and target [0, I:? How many of these are onto and how many of these are one-to-one? 5. f is onto and one-to-one. f is onto but not one-to-one. f is one-to-one but not onto. (iv) / is neither one-to-one nor onto. (v) Every integer is the image of exactly two integers.

Thus I cannot be one-to-one. This principle has far-reaching applications in combinatorics. Its name derives from the following flightful application: If more than n pigeons fly into n pigeonholes, then some pigeonhole must contain at least two pigeons. 3. Let S be a set of 11 or more binary numbers. Then at least 2 elements of S must have the same last digit when expressed in decimal notation. • 9}, where for s in SJ*(s) equals the last digit of s when s is expressed in decimal notation. Since lSI> 10 = ITI, there must be two numbers in S that map to the same element of T.

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