Week 12 Revision Discrete Math May 14, 2020 Marie Demlova: - - PowerPoint PPT Presentation

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Week 12 Revision

Discrete Math May 14, 2020

Marie Demlova: Discrete Math

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Example 1 Let A = {0, 1, 2, 3, 4, 5}. Define a mapping f : A → A is defined by f (x) = k, where k ≡ 4x ( mod 6). Decide whether f is one-to-one (injective), and whether f is onto (surjective). Example 2 Let f : X → Y and g : Y → Z be two mappings Decide whether the following statement is true or false: If f and g are one-to-one then so is g ◦ f .

Marie Demlova: Discrete Math

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Example 3 Give examples of sets A and B such that they are infinite sets with the same cardinality. Example 4 Let A = {(p, q) | p is an even integer, and q is an odd integer}. Decide whether A is a countable set. If yes, give the reasons.

Marie Demlova: Discrete Math

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Example 5 Given a relation R on the set of all natural numbers N by m R n iff m − n is divisible by 3 and by 5.

• 1. Decide whether R is reflexive (define a reflexive relation).
• 2. Decide whether R is symmetric (define a symmetric relation).
• 3. Decide whether R is antisymmetric (define an antisymmetric

relation).

• 4. Decide whether R is transitive (define a transitive relation).

Marie Demlova: Discrete Math

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Example 6 By mathematical induction prove that for every natural number n ≥ 1 it holds that 6 · 7n + 2 · 3n is divisible by 4. Example 7 By mathematical induction prove that for every natural number n ≥ 1 it holds that 2n+1 < 1 + (n + 1) 2n.

Marie Demlova: Discrete Math

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Example 8 Find all pairs of integers x and y for which 319 x + 473 y = 0. Example 9 Find all the pairs of integers x and y for which 10 x − 15 y = 131.

Marie Demlova: Discrete Math

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Example 10 Find the remainder when you divide 4254 + 2 · 17123 − 3 · 13102. by 15. Example 11 In (Z153, +, ·) an equation is given 14580(x + 1) = 22 − 8x. Find all its solutions.

Marie Demlova: Discrete Math

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Example 12 An operation ∗ is defined on the set Z × Z, i.e. the set containing all pairs of integers by: (u, v) ∗ (x, y) = (u + x, v · y).

• 1. Show that the pair (Z × Z, ∗) forms a semigroup. Write down

what is a semigroup.

• 2. Find a neutral element of the semigroup (Z × Z, ∗). Write

down what is a neutral element.

• 3. Find all invertible elements of the monoid (Z × Z, ∗). Write

down what is an invertible element.

• 4. If (Z × Z, ∗) is not a group find an equation that does not

have a solution. Is there an equation that has more than one solution? Justify your answers.

Marie Demlova: Discrete Math

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Example 13 Given the group of invertible elements of (Z22, ·). Decide whether it is a cyclic group. Yes or no does not suffice, you have to justify your answer. Example 14 Given the group of invertible elements (Z⋆

22, ·). How many

elements a subgroup of (Z⋆

22, ·) can have? For each such number

find a subgroup with this number of elements.

Marie Demlova: Discrete Math

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Example 15 The characteristic equation of a homogeneous linear difference equation of order 3 has roots λ1 = 1, and λ2,3 = −2 (i.e. the root λ = −2 has multiplicity 2). What is the general solution of this homogeneous linear difference equation? Example 16 Solve the following difference equation an+2 − 3 an+1 + 2 an = 6 · 2n, a0 = 2, a1 = 6.

Marie Demlova: Discrete Math

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Example 16 Given functions 4n, en, ln(n!), ln ln n, n lg n, ln(nln n), n, √n, 1 n. Sort these functions in f1, f2, . . . , such that fi ∈ O(fi+1) where i = 1, 2, . . . State all the pair of functions f , g for which f ∈ Θ(g). Example 17 Prove the following assertion: Given three non negative functions f (n), g(n), and h(n) for which there exists n0 ∈ N such that for every n ≥ n0 g(n) ≤ f (n) ≤ h(n). Assume that g(n) ∈ Ω(k(n)) and h(n) ∈ O(k(n)). Then f (n) ∈ Θ(k(n)).

Marie Demlova: Discrete Math