{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

2nd-07 - Discrete Mathematical Structures Second Mid—Term...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Discrete Mathematical Structures Second Mid—Term Exam 10/12/07 . n n2 n+ 1 2 PWJTI (1) Prove that Zk=1 k3 = ——(—4—2— for all n 21. --\_2 W‘- 0 1/ g (2) Prove that if A, B, and C are sets then, _ 0/“ ' (AUB)><C=(A><C)U(B><C) 621\$? If you interchange “U” and “X”, is the resulting equality true? Justify your answer. (3) The relations p and q are deﬁned on the set A— -— {2, 3, 4} as . follows: Mr z p = {(22,110 | 2:,y E A, x divides y}, 9” q=ﬁ\$mlayEAr+mhowl SEny (a ) List the elements of p and q and draw their directed graphs ) (b Find the matrices of p and q and obtain the relatiﬁi pg. (0) Find the transitive closure of q. 1’ \T \1 :3 (4 ) Let f- X —> Y and g- Y —> Z be functions. ‘ fa” ’ 7': } 2n) function. (b) In (a), give an example to show that 9 need not be inj ective. (5) Solve the recurrence relation S(lc)—S(k—1)—2S(k—2) = 216+ 1/ ﬂew! {152/1 with initial values 8(0) = 1, 3(1) = 1. m9! (Hint: Use alt + b as your trial solution)_ (ffa'¢.;1r£1n(€ ['1' {411-5145 .’ ~ 1 8715-? (a ) If g o f 1s an injective function prove that f 1s an injective it an... 4...__....,... ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online