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Discrete mathematics

Students are required to create 6-character long passwords to access the library. The letters must be from lowercase letters or digits. Each password must contain at least two lowercase-letters, at least one digit and contains no repeated digits. How many valid passwords are there? You are required to show your work step-by-step. [9 marks]

Note: 1hfgt1 is invalid because 1 appears more than once. 134ggg is valid because there are 3lower-case letters and digits are not repeated.

4. Consider the following automaton.

a) Give an example of a string containing 11 that is accepted by the following automaton. [2 marks]

b) Give an example of a string of length 8 that is rejected by the following automaton. [2 marks]

C) Describe the language of this automaton in plain English. [4 marks]

d) Describe the language of this automaton using Regular expression. [3 marks]

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Discrete mathematics

\text { 5. Given } R=\left(0^{*} 10^{\prime}\right)^{\prime} 1^{*} \text { and } S=\left(1^{*} 01\right)^{*}

a) Give an example of a string that is neither in the language of R nor in S. [2marks]

b) Give anexample of a string that is in the language of S but not R. [2 marks]

c) Give an example of a string that is in the language of R but not 5. [2 marks]

d) Give an example of a string that is in the language of Rand S. [2 marks]

e) Design a regular expression that accepts the language of all binary strings with no occurrences of b ab [4 marks]

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Discrete mathematics

Prove the following statement by induction. For all nonnegative integers nn, 3 divides n^3 +2n+3n3+2n+3. State the mathematical induction and show your work clearly.

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Discrete mathematics

4. Consider the triangle below

Suppose angle B has pi/6measureradians. Determine the area of Triangle ABC if angle A is acute.Show ell work

Show all work.

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Discrete mathematics

3. Determine side length BC in Triangle 3, showing all work.

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Discrete mathematics

2. For Triangle 1, calculate the measure of angle N. Then explain why it is impossible for N to actually bean obtuse angle. Show all of your work.

2. For Triangle 1, calculate the measure of angle N. Then explain why it is impossible for N to actually bean obtuse angle. Show all of your work.

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Discrete mathematics

1. For each of the triangles above, Fill in the blanks below with "LoC" (Law of Cosines), "LOS" (Law ofSines) or "either" depending on which Law you'd use first to solve for missing sides or angles. Then give a brief 1 sentence justification for your choice.

a. In Triangle 1I would usefirst, because

b. In Triangle 2 1 would use________first, because

c. In Triangle 3 I would use_____first, because

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Discrete mathematics

(4) Is V4 irrational? What goes wrong if we try to implement the proof that we did for irrationality of V2?

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Discrete mathematics

(3) If k is an odd integer and m is an even integer, then k? + m² is odd.

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Discrete mathematics

(2) Recall that lim→a f(k) = L means

For all real numbers e > 0 there exists a real number d > 0 such that for all real numbers x if a – 8 < x < a + d and x # a, then L – e < f(x)< L +e

Negate this statement.

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