Department of Mathematics and Philosophy of Engineering MHZ4256 Mathematics For Computing Assignment No.01 [PDF]

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Department of Mathematics and Philosophy of Engineering MHZ4256 Mathematics for Computing Assignment No.01 Due Date:



Will be notify later



Academic Year: 2020/2021



Instructions •



Answer all the questions.



•



Attach the cover page with your answer scripts.



•



Use both sides of papers when you are answering the assignment.



•



Write your address back of your answer scripts.



Q1. a) Let, P: We should be honest. Q: We should be dedicated. R: We should be overconfident. Write logical expression for ‘We should be honest or dedicated but not overconfident'.



b) i.



Consider the statement "𝑥 > 0 ⇒ 𝑥 + 1 > 0". Is this statement true or false?



ii.



Show that (𝑃 → 𝑄) ∨ (Q → P) is a tautology.



i.



Make a truth table for the statement ¬𝑃 ∧ (𝑄 → 𝑃). what can you conclude about



c)



𝑃 and 𝑄 if you know the statement is true? ii.



Determine whether the following two statements are logically equivalent. ¬(𝑃 → 𝑄)𝑎𝑛𝑑 𝑃 ∧ ¬𝑄



d) State the “converse”, “inverse”, and “Contrapositive” of each of the following statements. i.



If two angles are congruent, then they have the same measure.



ii.



If (5𝑥 – 1 ) = 9, then 𝑥 = 2



Q2. a) Let, 𝑃(𝑥): “𝑥 knows Kung Fu” 𝑄(𝑥): “𝑥 knows Karate” Where the domain consist of all adults in your neighborhood. Write the following sentence using predicates and quantifiers. i.



There is an adult in your neighborhood who knows Kung Fu and Karate.



ii.



Every adult in your neighborhood knows Kung Fu or Karate.



iii.



No adults in your neighborhood knows Kung Fu or Karate



b) Test the validity of following argument: If I want to be a lawyer, then I want to study logic. If I don’t want to be a lawyer, then I don’t like to argue. .………………………………………………………… Therefore, if I like to argue, then I want to study logic.



c) Using an appropriate method, prove the followings: i.



For all positive integers 𝑛, 𝑛



∑ 𝑟(𝑟 + 1) = 𝑟=1



𝑛(𝑛 + 1)(𝑛 + 2) 3



ii.



For any positive integer 𝑛, 6𝑛 − 1 is divisible by 5.



iii.



Consider two integers 𝑎 and 𝑏. If 𝑎𝑏 is an even number then 𝑎 or 𝑏 is even.



Q3. a) A marketing survey of 1,000 commuters found that 600 answered that they listen to the news, 500 listen to music, and 300 listen to both. Let N = set of commuters in the sample who listen to news and M = set of commuters in the sample who listen to music. Fill out a two-set Venn diagram and give the number in each of the sets below. i.



𝑁∩𝑀



ii.



𝑁′ ∩ 𝑀



iii.



𝑁 ∪ 𝑀′



b) If 𝐴 = { 𝑥 | 𝑥 ≤ 6 ; 𝑥 ∈ ℕ} and 𝐵 = {𝑥 |3 ≤ 𝑥 < 9, 𝑥 ∈ ℕ}, find, i.



𝐴 ∪ 𝐵



ii.



𝐴– 𝐵



c) Prove the following using the set theory laws, i.



A − B = B ′ − A′



ii.



𝐴 – (𝐴 ∩ 𝐵) = 𝐴 – 𝐵



d) Show that A ∪ (B − A) = A ∩ B.



Q4. a) Define the relation by the usual notation. b) Let 𝐴 = {1, 2, 3, 4} and 𝐵 = {5, 7, 9}. Determine, i.



𝐴 × 𝐵



ii.



𝐵 × 𝐴



iii.



Is 𝐴 × 𝐵 = 𝐵 × 𝐴 ?



iv.



Is 𝑛(𝐴 × 𝐵) = 𝑛 (𝐵 × 𝐴) ?



c) Find the domain and range of the relation R given by, 𝑅 = {(𝑥, 𝑦) ∶ 𝑦 = 𝑥 + 6𝑥 ; where 𝑥, 𝑦 ∈ ℕ and 𝑥 < 6}. d) Which of the following relations are functions from 𝐴 to 𝐵. Write their domain and range. If it is not a function give reasons. 𝐴



𝐵



𝑅



i.



𝐴 = {1,3,4,8}



𝐵 = {−2,7, −6,1,2}



𝑅1 = {(1, −2), (3,7), (4, −6), (8,1)}



ii.



𝐴 = {1,2,4}



𝐵 = {0,7, −1,3,10}



𝑅2 = {(1,0), (1, −1), (2,3), (4,10)}



Q5. a) Let 𝑅 be a relation on set {1,2,3,4} with, 𝑅 = {(1,1), (1,4), (2,3), (3,1), (3,4)} Find the reflexive, symmetric and transitive closure of 𝑅. b) Show that the relation R is an equivalence relation in the set 𝐴 = { 1, 2, 3, 4, 5 } given by the relation, 𝑅 = { (𝑎, 𝑏): |𝑎 − 𝑏| 𝑖𝑠 𝑒𝑣𝑒𝑛 }



c) Let, 𝑓(𝑥) = √1 − 𝑥



and



𝑔(𝑥) = 𝑥 2 − 3



Find the composition of functions and its domain. d) Classify each function as injective, surjective, bijective or none of these. i.



𝑓: 𝑁 → 𝑁 defined by 𝑓(𝑛) = 𝑛 + 3.



ii.



𝑓: 𝑅 → 𝑅 defined by 𝑓(𝑥) = 𝑥 3 − 𝑥.



iii.



𝑓: 𝑍 → 𝑍 defined by 𝑓(𝑛) = 𝑛 − 5.



iv.



𝑓: 𝑁 → 𝑄 defined by 𝑓(𝑛) = 1/𝑛.



v.



𝑓: 𝑁 → 𝑍 defined by 𝑓(𝑛) = 𝑛2 − 𝑛.