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Discrete_Mathematics_Assignment__5

Discrete_Mathematics_Assignment__5 - Gunjan Patel...

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Gunjan Patel | --------------------------------------------------------------------------------------------------- ------------------ Discrete Mathematics – Assignment # 5 --------------------------------------------------------------------------------------------------- ------------------ . ( ): |7 1 Recurrence Relations page 457 Exercise 8| Find the solution to each of these recurrence relations with the . given initial conditions Use an iterative approach such as that . used in Example 5 ) a = = . . . = = --------------------------------------------------------------------------------------------------- ------------------ ) b = = . . . = = --------------------------------------------------------------------------------------------------- ------------------ ( ) c = = . . . = - ( +( - ) + ( - ) + ... + ( -( - ))) + n n 1 n 2 n n 1 = ( ) d = = . . . =
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Gunjan Patel | = --------------------------------------------------------------------------------------------------- ------------------ ( ) e ... = ( + )!* = ( + )! n 1 a0 2 n 1 ( ) f = ( ( - )) 2n 2 n 1 . . . = = * 3 2 n ! --------------------------------------------------------------------------------- --------------------------------------
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Gunjan Patel | . - ( ): 7 5 Inclusion Exclusion page 505 Exercise 20| How many elements are in the union of five sets if the sets , , contain 10 000 elements each each pair of sets has 1000 , common elements each triple of sets has 100 common , , elements every four of the sets have 10 common elements and there is I element in all five sets?
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Gunjan Patel | Using the Inclusion-Exclusion Principle: |A| + |A | + |A| + |B = (10000 + 10000 + 10000 + 10000 + 10000) – (1000 -1000 -1000 -1000 -1000 -1000 -1000 -1000 -1000 -1000) + (100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100) – (10 – 10 – 10 – 10 – 10) + 1 = ( * ) – ( * ) + ( * ) – ( * ) + r 10000 5 1000 10 100 10 10 5 1 = + + = 50000 10000 1000 50 1 40951 --------------------------------------------------------------------------------------------------- ------------------ . ( ): |8 5 Equivalence Relations page 565 Exercise 48| List the ordered pairs in the equivalence relations produced by these partitions of { , , a b , e , , d e , f }. g ( ) a { , }, { , }, { , a b c d e , f } g {( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , a a a b b a b b c c c d d c
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