Discrete Structures Test #2 SII 2001

1) Draw a line to connect each equivalent statement:

a) (A Ç B) È Æ A È A^c

b) A Ç A^c A - B^c

c) A - B Æ

d) U A Ç B^c

2) Reduce A È (B Ç C^c )^c È C :

3) List all the elements in the power set of B = {0, 1}:

a)

b) What is the cardinality of P(A), where |A| = 7.

4) Determine , a_-3 = 1, a_-2 = 0, a_-1 = 0, a₀ = 0, a₁ = 0, a₂ = 1, a₃ = 1, a₄ = 1.

5) T/F

a) {4} Í {{2}, {4}, {6}}

b) 4 Î {n | n = 2k, "k Î Z}

c) 4 Î {{2}, {4}, {6}}

d) {4} Î P({1, 2, 3, 4})

e) {4} Í P({1, 2, 3, 4})

f) {4} Í {2, 4, 6}

6) Write out the symbolic representation of the Quotient-Remainder Theorem. Given any integer n and any positive integer d, there exist integers q and r such that:

7) Use the geometric series formula to find an explicit formula:

8) List the first three terms of the sequence c_i = (-1) ⁱ⁺¹ / 2 ⁱ⁺¹, for all integers i ³ 0:

9) Given the Java integer: 1100 0010 0000 0011 1000 0001 1001 0100

a) Hexadecimal Representation:

b) Decimal Value:

10) Convert 188.09375₁₀ to octal and to binary:

11) Consider the Java bytes A = 1010 1010, B = 0101 0101. Convert their sum,
A + B, to decimal:

12) The sum of the first 150 integers is:

13) Prove: The product of an even integer and an odd integer is even.

14) Use mathematical induction to prove that 1 + 3 + 5 + … + (2n - 1) = n², for all integers n ³ 1.

15) Circle the letter corresponding to any of the following relations that are Second Order Linear Homogeneous with Constant Coefficients:

a) a_k = (k - 1)a_{k - 1} + 2a_{k - 2}

b) b_k = 8b_{k - 1} - 16b_{k - 2}

c) c_k = 3c_{k - 2} + 1

d) d_k = 4d_{k - 2}

16) Pick one of the SOLHCC recurrence relations from problem 15 and determine its explicit formula given that term₀ = 1 and term₁ = 2.
Characteristic equation: t² - At - B = 0. Distinct Roots Theorem: a_n = Crⁿ + Dsⁿ. Single Root Theorem: a_n = Crⁿ + Dnrⁿ. Quadratic equation: .

a) Explicit Formula:

b) term₂ =

c) term₃ =