Assignment 4
Instruction. Solve the following problems. Assume any data if necessary.
1. Obtain the product of sums canonical form of the Boolean expression
f (x, y, z) = x + yz
2. Simplify the following Boolean expressions
(a) x + x(x + y) + yz
(b) a + ab
(c) (x + y)(x + y)
3. Draw a logic circuit with ((x ∧ y) ∨ z)) as output.
4. A binary operation ∗ is defined on Z by a ∗ b = a + b + 2, a, b ∈ Z. Show that (Z, ∗)
is a group.
5. Let A = {1, 2, 3, 4, 5, 6, 7} and a relation R is defined as
R = {(x, y) | |x − y| = 2}.
Is R a transitive relation ?
6. Show that for any ring R with unity, (R, +) is an Abelian group.
7. Show that (Q+ , ∗) is an Abelian group, where a ∗ b = ab/2, for all a, b ∈ Q+ .
8. Find the solution of the following recurrence relations
(a) sk − 3sk−1 − 4sk−2 = 4k .
√
(b) an + 5an−1 + 5an−2 = 0, with a0 = 0, a1 = 2 5.
(c) an = 3an−1 + 2, a0 = 1 (use generating functions).
9. Find the coefficient of x10 in (1 + x)−5 .
10. Using mathematical induction prove that
1 + 4 + 9 + 16 + · · · + n2 =
1
n(n + 1)(2n + 1)
.
6
11. From the textbook titled Graphs1 in Moodle
(a) pg 665, questions 5, 26, 55
(b) pg 675, questions 9, 15, 29, 30, 35, 39, 43
(c) pg 705, questions 5, 7, 26, 31, 35
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