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Discrete Mathematics Final Exam
Comprehensive Quiz Study Guide with Answers
Discrete Math for Computer Science | 100% Pass Guaranteed | Graded A+ |
DISCRETE MATH Exam
Course Title and Number: DISCRETE MATH Exam Title: DISCRETE MATH Midterm and Final Exam Date: Midterm and Final Exam 2024- 2025 Instructor: [Insert Instructor’s Name] Student Name: [Insert Student’s Name] Student ID: [Insert Student ID]
Examination
180 minutes
Instructions:
**1. Read each question carefully.
- Answer all questions.
- Use the provided answer sheet to mark your** **responses.
- Ensure all answers are final before submitting the** **exam.
- Please answer each question below and click Submit** **when you have completed the Exam.
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- This is Exam which will assess your knowledge on** the course Learning Resources.
Good Luck!
Read All Instructions Carefully and Answer All the Questions Correctly Good Luck: - Conditional Statements - Answer>> P => Q Converse of a Conditional Statement - Answer>> Q => P Contrapositive of a Conditional Statement - Answer>> -Q => -P Equivalent Statements - Answer>> Have the same logical value (Wether T or F) Compound Statements - Answer>> a statement formed by using a connective to join two independent clauses or logical statements. Tautology - Answer>> a formula which is always true for every value of its propositional variables Statement - Answer>> A statement (proposition) is a sentence that is either true or false, but not both. Negation - Answer>> A negation of a statement p is the statement "not p" or "it is not the case that p," and is denoted by ⌐p. A statement and its negation always have the opposite truth value. Conjunction - Answer>> The conjunction of two statements p and q is the statement "p and q," and is denoted by p ᴧ q. The conjunction p ᴧ q is true if both p and q are true and is false otherwise. Disjunction - Answer>> The disjunction of two statements p and q is the statement "p or q," and is
Distributive Laws - Answer>> p ᴧ (q v r) ≡ (p ᴧ q) v (p ᴧ r) p v (q ᴧ r) ≡ (p v q) ᴧ (p v r) Impotent Laws - Answer>> p ᴧ p ≡ p p v p ≡ p Double Negation Law - Answer>> ⌐(⌐p) ≡ p Negation Laws - Answer>> p ^ ⌐p ≡ F p v ⌐p ≡ T Identity Laws - Answer>> p ᴧ T ≡ p p v F ≡ p Universal Bound Laws - Answer>> p ᴧ F ≡ F p v T ≡ T Absorption Laws - Answer>> P ᴧ ( p v q) ≡ p p v (p ᴧ q) ≡ p Variations of the Conditional Statement - Answer>> Conditional p→ q Converse q → p Inverse ⌐p → ⌐q Contrapositive ⌐q → ⌐p Negation of Conditional - Answer>> ⌐( p → q) ≡ p ᴧ ⌐ q Universe of Discourse - Answer>> The set of possible values Universal Quantification - Answer>> The universal quantification of P(x) is the statement " For all x in the universe of discourse, P(x) is true." It is denoted by
¥ x P(x) Existential Quantification - Answer>> The existential quantification of P(x) is the statement "There exists an x in the universe of discourse for which P(x) is true." It is denoted by Ǝ x P(x) Negations of Quantified Statements - Answer>> For any P(x) ⌐¥ x P(x)= Ǝ x ⌐P(x) ⌐Ǝ x P(x) = ¥ x ⌐P(x) Argument - Answer>> An argument is a series of statements, called PREMISES, followed by another statement, called a CONCLUSION. An argument is called VALID if the conclusion is true whenever all the premises are true. Standard Valid Arguments - Answer>> Modus Ponents, Modus Tollens, Reasoning by Transitivity(Law of Hypothetical Syllogism), Disjunctive Syllogism Standard Invalid Arguments - Answer>> Fallacy of the Converse, Fallacy of the inverse Modus Ponens - Answer>> p → q p
Therefore q Modus Tollens - Answer>> p → q ⌐q
Therefore ⌐p Reasoning by Transitivity(Law of Hypothetical Syllogism) - Answer>> p → q
Contradiction - Answer>> To prove a statement p, start by assuming the statement is false. In other words assume ⌐p. Then show that assuming ⌐p leads to a contradiction. Set - Answer>> A set is an unordered collection of objects. The objects in the set are called the ELEMENTS of the set. If x is an element of a set S, we denote this by x Є S. If x is not an element of a set S, we denote this by x Ɇ S. Empty Set - Answer>> The empty set is the set with no elements, and it is denoted by Ø. It may also be denoted by { }. Equal Sets - Answer>> Two sets A and B are said to be equal if they have the same elements. In this case we write A = B Subset - Answer>> A set B is said to be a subset of a set A if every element of B is also an element of A. In other words, x Є B implies x Є A. To denote that B is a subset of A, we write B C A Cardinality - Answer>> If A is the finite set, then the cardinality of A is the number of distinct elements in A, and is denoted by |A|. Cartesian Product - Answer>> The Cartesian Product of two sets A and B is the set of all ordered pairs (a,b), where a Є A and b Є B. Is denoted by A x B. Therefore, A x B = { (a,b) | a Є A and b Є B}. Power Set - Answer>> For any set S, the power set of S is the set whose elements are all the subsets of S, and it is denoted by P(S). in other words,
P(S) = {B | B c S}. Union - Answer>> The union of two sets A and B is the set whose elements are elements of A, B, or both. It is denoted by A U B. Therefore, A U B= {x | x Є A or x Є B}. Intersection - Answer>> The intersection of two sets A and B is the set whose elements are elements of both A and B. It is denoted by A ∩ B. Therefore, A ∩ B = { x | x Є A and x Є B}. Difference - Answer>> Let A and B be sets. The diffence of A and B is the set whose elements are elements of A but not elements of B. It is denoted by A
- B. Therefore, A - B = { x |x Є A and x Є B }. Complement - Answer>> The complement of a set A is the set U - A whose elements are the elements of the universal set U that are not elements of A. It is denoted by Ā. Therefore, Ā = { x | x Ɇ A }. Divides - Answer>> If a and b are integers and there is an integer c such that b = a • c, then we say a divides b or b is divisible by a, and write a|b. In this case, we say that a is a factor or divisor of b and that b is a multiple of a. If a does not divide b, we write a ł b. Properties of Divisibility - Answer>> For all integers a, b and c, the following properties hold.
- If a | b and b | c, then a | c
- if a | b and a | c, then a | ( b + c)
- if a | b, then a | bc
a modulo n (a mod n) - Answer>> If n is a positive integer and a is any integer, then the remainder we get when using the division algorithm to divide a by n is calld a modulo n. and is denoted a mod n Congruent modulo n - Answer>> Let n be a positive integer. Two integers a and b are said to be congruent modulo n if n | (b - a). We write a ≡ b (mod n) to denote a is congruent to b modulo n, and write a ≠ b (mod n) to indicate that this is not the case. Congruence Modulo n. - Answer>> Let n be a positive integer, and a and b any integers. Then a ≡ b (mod n) if and only if the leave the same remainder when divided by n using the division algorithm, that is if and only if a mod n = b mod n. Properties of Modular Arithmetic - Answer>> Let a, b, c, d, and n be integers with n > 1, and suppose a ≡ b (mod n) and c ≡ d (mod n). Then,
- (a + c) ≡ (b + d) ( mod n)
- (a - c) ≡ ( b - d) ( mod n)
- ac ≡ bd (mod n) Inverse of a modulo n. - Answer>> If a is an integer and ā a ≡ 1( mod n), then we say ā is an inverse of a modulo n. Chinese Remainder Theorem - Answer>> Suppose n1,n2,....,nk are integers which are pairwise relatively prime. Then for any integers a1, a2,...,ak, there is a solution x to the system of equations x ≡ a1 (mod n1) x≡ a2(mod n2)....... x≡ ak(mod nk)
x = a1• N1•x1 + a2•N1•xd mod N where, N = n1•n N1= N/n N2=N/n x1= inverse of N1 mod n x2 = inverse of N2 mod n Function - Answer>> Let A and B be sets. A function f from A to B associates each element of A with a unique element of B. More formally, ever a Є A is associated with a unique element f(a) = b Є B called the image of a under f. We say that f MAPS a to f(a). This function sometimes denoted by f: A → B Domain - Answer>> Let f : A → B be a function from A to B. We call A the domain of f Codomain - Answer>> Let f : A → B be a function from A to B. The set B is called the codomain of f. Range - Answer>> Let f : A → B be a function from A to B. The set of all images of the elements of A under the function f is called the range of f. in other words range of f = { b Є B | b = f(a) for some a Є A }. one-to-one - Answer>> A function f : A → B from A to B is called one-to-one if f (a1) = f (a2) implies a1 = a2 for all a1, a2 Є A. Onto - Answer>> A function f : A → B from A to B is called onto if every element of B is an image of some element of A under f. That is, f is onto if given any b Є B, there exists some element a Є A such that f (a) = b
Cardinality - Answer>> If A and B are 2 sets, then we say A and B have THE SAME cardinality if there exists a map f : A → B that is a bijection. Larger Cardinality - Answer>> We say that set A has LARGER cardinality than B if there exists a one-to-one function g : B → A. and there does not exists a bijection h : B → A. Countable - Answer>> A set A is called countable(countably infinite) if there exists a bijection f : { 1,2,3,....} → A (i.e. - we can make an infinite lists of all elements of A.) Contradiction - Answer>> a proof technique used by assuming that a statement is false and then showing that a contradiction follows on that assumption. Universally Quantified Statements - Answer>> ∀ - denotes "for all". Existentially Quantified Statements - Answer>> ∃ - denotes "there exists" Conjunction - Answer>> ^ - denotes "AND" Disjunction - Answer>> ∨ - denotes "OR" Exclusive OR - Answer>> ⊕ - denotes "XOR" Commutative Laws - Answer>> 𝑝∨𝑞⇔𝑞∨𝑝 and 𝑝∧𝑞⇔𝑞∧𝑝 Associative Laws - Answer>> (𝑝∨𝑞)∨𝑟⇔𝑝∨(𝑞∨𝑟)
and (𝑝∧𝑞)∧𝑟⇔𝑝∧(𝑞∧𝑟) Distributive Law - Answer>> 𝑝∧(𝑞∨𝑟)⇔(𝑝∧𝑞)∨(𝑝∧𝑟) and 𝑝∨(𝑞∧𝑟)⇔(𝑝∨𝑞)∧(𝑝∨𝑟) Identity Law - Answer>> 𝑝∨ 0 ⇔𝑝 and 𝑝∧ 1 ⇔𝑝 Negation Laws - Answer>> 𝑝∧¬𝑝⇔ 0 and 𝑝∨¬𝑝⇔ 1 Absorption Laws - Answer>> 𝑝∧(𝑝∨𝑞)⇔𝑝 and 𝑝∨(𝑝∧𝑞)⇔𝑝 DeMorgan's Laws - Answer>> ¬(𝑝∨𝑞)⇔(¬𝑝)∧(¬𝑞) and ¬(𝑝∧𝑞)⇔(¬𝑝)∨(¬𝑞) Involution Laws - Answer>> ¬(¬𝑝)⇔𝑝 Subset Inclusion - Answer>> The relationship of one set being a subset of another. A⊂B Set Equality - Answer>> When all of the items in two or more sets are the same and the number of elements is also the same. A=B
The set Zn - Answer>> the set of all equivalence classes of integers modulo n Zn = {[x]n | x in Z} (if n ≠ 0 then Zn has exactly n elements) Euler's Theorem - Answer>> Let n in N be given. For each [a]n in Zn [a]n^φ(n) = [1]n Eulers Totient Function - Answer>> φ: N => N : φ(n) = #Zn (basically φ(n) is equal to the amount of numbers that care coprime to n) Using Eulers Theorem to reduce power towers - Answer>> if we have [2023^2024]12 then φ(12) = [2024]4 = 0 so [2023^0]12 = [1]12 = Bezout's Theorem and Extended Euclidean algorithm - Answer>> States that given x0,x1 in Z, there must exist integers a,b in Z that satisfies ax0 + bx1 = gcd(x0,x1) The Extended Euclidean algorithm returns the exact values of the integers a,b Supposing x0>x Extended Euclidean algorithm proceeds by recursivly defining x2 = x0 mod x1 = x0 - (the amount of x1s that fit into x0s)x x3 = x1 mod x2 = x1 - (the amount of x2s that fit into x1s)x until one of the x(k+1) = 0 the value xk (the value before it equals zero is always equal to the gcd of x0 and x1 so you must solve for xk using a combination of x's of lower index
Now if we suppose that x0 = n and x1 = w such that [w]n is in Z*n we have the gcd(x0,x1) = 1 applying mod n [a]n * [n]n + [b]n * [w]n = [1]n since [n]n = 0 [b]n * [w]n = [1]n which means [b]n = [1]n/[w]n Equivalence Relation - Answer>> reflexive, symmetric, and transitive. model the = sign Pre-Order - Answer>> reflexive and transitive. Partial Order - Answer>> reflexive, anti-symmetric, and transitive. model the <= sign Strict Order - Answer>> anti-reflexive and transitive model the < sign reflexive - Answer>> for all x in X : xRx symmetric - Answer>> for all x,y in X : xRy => yRx transitive - Answer>> for all x,y,z in X : xRy ^ yRz => xRz anti-reflexive - Answer>> for all x in X : ¬(xRx) anti-symmetric - Answer>> for all x,y in X : xRy ^ yRx => x = y
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