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# TCSS 321 Discrete Mathematics

0 Download 5 Pages / 1,210 Words 04-06-2021
• Course Code: TCSS321
• University: University Of Washington
• Country: United States

## Question:

1. Recurrences[10%] Consider the recurrence T(n). [Bonus Question]
1, if n = 1
5, if n = 2
5T(n − 1) − 6T(n − 2), if n > 2 & n ∈ N.

Prove by strong induction that T(n) = 3n −2 for all n ≥ 1 and n ∈ N.2. Mathematical Induction[10%]. Prove using mathematical induction that 3 divides n 3 + 2n for every n ∈ Z +.
3. Relations [10%]. Find a relation R on the set Z that is
• Not reflexive.
• Not symmetric.
• Not transitive.

5. Relations [10%]. Let S = {{1}, {1, 2}, {1, 3}} be a set. Is there exists a totally ordered binary relation R on A. Prove your claim. Relations [10%] Define a binary relation R on Z as (x, y) ∈ R only if
| x − y |< 1.
• Is R Reflexive. Prove or disprove your claim.
• Is R Symmetric. Prove or disprove your claim.
• Is R Transitive. Prove or disprove your claim.
• Is R Antisymmetric. Prove or disprove your claim.
• Is R Comparable. Prove or disprove your claim.

6. Set Partition [10%] Let Z3 = {[0], [1], [2]} denote the set of equivalence classes modulo 3. Prove that [0] ∩ [1] = ∅, [0] ∩ [2] = ∅, and [1] ∩ [2] = φ.7. Number Theory[10%]. Prove that if n ∈ Z then n 2 ≡ 0 (mod 4) or

Number Theory[10%]. Prove that if a, b, c and m are integers such that m ≥ 2, c > 0, and a ≡ b (mod m), then ac ≡ bc (mod mc). Division Theorem [10%]. Prove that every prime number except 2 and 3 is of form 6k + 1 or 6k + 5 for some k ∈ Z. Hint note n = 6 and apply GCD [10%]. Let a, b, d ∈ Z such that d = GCD(a, b). Let d 0 ∈ Z be .

11. Number Theory[10%]. Prove that the fundamental theorem of arithmetic cannot be extended to negative integers.
12.CRT [10%] Solve the following system of congruences. [Bonus Question]
2x ≡ 5 (mod 7)
4x ≡ 2 (mod 5)
3x ≡ 9 (mod 11)

13. Fermat’s Little Theorem [10%]. Using Fermat’s little theorem, prove that if n is a positive integer, then 42 divides n Quadratic Residusity [10%]. Let p be an odd prime and 1 ≤ a ≤ p−1. Prove that if a ∈ QRp is a quadratic residue modulo p then a p−1 2 ≡ 1 (mod p).

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