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Mathematics for Natural Sciences, Summaries of Mathematics

Addis Ababa UniversityMathematics
Prof. Ayele Mandefro Mulugeta

In this chapter, we study the basic concepts of propositional logic and some part of set theory. In the first part, we deal about propositional logic, logical connectives, quantifiers and arguments. In the second part, we turn our attention to set theory and discus about description of sets and operations of sets.

Typology: Summaries

2020/2021

Uploaded on 03/21/2024

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ADDIS ABABA UNIVERSITY
DEPARTMENT OF MATHEMATICS
Mathematics for Natural Sciences (Math 1011)
worksheet I - November 2019
Part I: Propositional Logic.
(1) Which of the following sentences are propositions? What are the truth values of those
that are propositions?
(a) 5 <9 and there are infinitely many prime numbers.
(b) x2=y2provided that x=y.
(c) How many courses are you taking this semester?
(d) The first human kind lived in Ethiopia.
(2) Consider the following propositions
r: Rabbits have been seen in the area.
b: Berries are ripe along the path w: Walking on the path is safe.
Write the following propositions using r,band wand logical connectives.
a) It is not safe to walk along the path, but rabbits have not been seen in the area
and the berries along the path are ripe.
b) For walking on the path to be safe, it is necessary but not sufficient that berries
not be ripe along the path and for rabbits not to have been seen in the area.
c) Walking is not safe on the path whenever rabbits have been seen in the area and
berries are ripe along the path.
(3) Given the following propositions
p: You have the flu. r: You pass the course.
q: You miss the final examination.
Express each of the following propositions as an ordinary English sentence.
a) ¬q⇔rb) ¬(p∧q) c) (p⇒ ¬r)∨(q⇒ ¬r).
(4) Determine the truth value of pif
a) (q⇒ ¬p)∨ris False b) ¬qand p⇒qare True
c) (p∨ ¬q)⇒(¬r∨p) is False.
(5) If ¬[¬r⇒ ¬(p∧q)] is true, then find the truth value of [(p⇔r)∨q]⇔(¬p∧r).
(6) Prove that
a) (p∧q)⇒r≡p⇒(q⇒r) b) (q∨r)⇒p≡(q⇒p)∧(r⇒p).
(7) For the following propositions, indicate whether it is a tautology, a contradiction, or
neither. Use a truth table to decide.
a) [p∧(p⇒q)] ⇒qb) (p⇒q)∧ ¬(¬q⇒ ¬p).
c) (¬q⇒ ¬p)⇒((¬q⇒p)⇒q).
(8) Determine whether the following statements are valid or invalid.
a) If I do not wake up, then I cannot go to work.
If I cannot go to work, then I will not get paid.
Therefore, if I do not wake up, then I will not get paid.
b) If I study, then I will not fail Math 1011.
If I do not play cards to often, then I will study.
I failed Math 1011.
Therefore, I played cards too often.
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pf2

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ADDIS ABABA UNIVERSITY

DEPARTMENT OF MATHEMATICS

Mathematics for Natural Sciences (Math 1011) worksheet I - November 2019

Part I: Propositional Logic.

(1) Which of the following sentences are propositions? What are the truth values of those that are propositions? (a) 5 < 9 and there are infinitely many prime numbers. (b) x^2 = y^2 provided that x = y. (c) How many courses are you taking this semester? (d) The first human kind lived in Ethiopia. (2) Consider the following propositions r : Rabbits have been seen in the area. b : Berries are ripe along the path w: Walking on the path is safe. Write the following propositions using r, b and w and logical connectives. a) It is not safe to walk along the path, but rabbits have not been seen in the area and the berries along the path are ripe. b) For walking on the path to be safe, it is necessary but not sufficient that berries not be ripe along the path and for rabbits not to have been seen in the area. c) Walking is not safe on the path whenever rabbits have been seen in the area and berries are ripe along the path. (3) Given the following propositions p : You have the flu. r: You pass the course. q : You miss the final examination. Express each of the following propositions as an ordinary English sentence. a) ¬q ⇔ r b) ¬(p ∧ q) c) (p ⇒ ¬r) ∨ (q ⇒ ¬r). (4) Determine the truth value of p if a) (q ⇒ ¬p) ∨ r is False b) ¬q and p ⇒ q are True c) (p ∨ ¬q) ⇒ (¬r ∨ p) is False. (5) If ¬[¬r ⇒ ¬(p ∧ q)] is true, then find the truth value of [(p ⇔ r) ∨ q] ⇔ (¬p ∧ r). (6) Prove that a) (p ∧ q) ⇒ r ≡ p ⇒ (q ⇒ r) b) (q ∨ r) ⇒ p ≡ (q ⇒ p) ∧ (r ⇒ p). (7) For the following propositions, indicate whether it is a tautology, a contradiction, or neither. Use a truth table to decide. a) [p ∧ (p ⇒ q)] ⇒ q b) (p ⇒ q) ∧ ¬(¬q ⇒ ¬p). c) (¬q ⇒ ¬p) ⇒ ((¬q ⇒ p) ⇒ q). (8) Determine whether the following statements are valid or invalid. a) If I do not wake up, then I cannot go to work. If I cannot go to work, then I will not get paid. Therefore, if I do not wake up, then I will not get paid. b) If I study, then I will not fail Math 1011. If I do not play cards to often, then I will study. I failed Math 1011. Therefore, I played cards too often. 1

(9) Let P (x) : x is an integer greater than 5 Q(x) : x is a natural number. Determine the truth value of the following propositions. a) P (3) ∧ Q(−2) b) P (2) ⇒ Q(2) c) [P (3) ⇒ Q(4)] ∨ Q(9). (10) Let P (x) : x is a prime number Q(x) : x is an even number R(x) : x is an odd number. S(x) : x is an integer. Write a sentence, which corresponds to each of the following: a) R(x) ∨ Q(x) ⇒ P (x) b) P (x) ⇔ Q(x) ∨ R(x). (11) If U = R, then find the truth value of (∀x)(∃y)(

x^2 = 16 ⇒ y + x = 10). (12) Let P (x) : x is a composite number Q(x) : x is a prime number. Find the truth values of (i) (∃x)[P (x) ⇔ Q(x)] (ii) (∀x)[P (x) ⇔ Q(x)]. (13) Find the truth values of the following where U = R a) (∃x)(∀y)(x^2 < y^2 ) b) (∀x)[x 6 = 0 ⇒ (∃y)(xy = 4)].

Part II: Set Theory.

(1) Write the following sets in complete listing or partial listing method. a) A = {x|x is an integer and 3 < x ≤ 10 } ∩ {x|x is even integer}. b) B = {x ∈ N|(x − 1)(x − 3) 6 = 0 ⇔ x + 1 = x}. c) C = {x ∈ N|x ≥ 3 ⇔ x < 0 }. d) D = {x|x ∈ N ∧ (x − 1)(x − 3) 6 = 0 ⇔ x + 1 = x}. (2) If n(A\B) = 18, n(A ∪ B) = 70 and n(A ∩ B) = 25, then find n(B). (3) For all subsets A, B and C of some Universal set U , prove or disprove the following statements. a) (A ⊆ B ∧ A ⊆ C) ⇒ A ⊆ B ∩ C b) A ⊆ B ⇔ A ∩ B = A c) Ac\Bc^ = B\A d) A ⊆ B ⇒ A ∪ (B\A) = B. (4) Let U = {x ∈ Z| − 12 ≤ x ≤ 6 }. A = {x ∈ Z|x = 2n ∧ x = 3m, for − 4 ≤ n ≤ 8 and − 2 ≤ m ≤ 3 }. B = { 0 , 1 , 3 } C = {x ∈ Z|x = n 3 , n ∈ A}. Then find a) A[B ∪ C]c^ b) (A\B)(A ∪ C)c^ c) (A ∩ B)\C. (5) (a) For every natural number n, define An = {x ∈ Z|− n^24 ≤ 2 x + 1 ≤ (^24) n }. Then find i) ∪^6 n=1An ii) [ ( ∪^4 n=1An) ( ∩^3 n=1An) ] c^ iii) ∩An, n ∈ N. (b) For every natural number n, define An = {x ∈ N|x = n ⇒ x < n}. Then find i) ∪^10 n=1An ii) ∪An, n ∈ N.

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