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1. faculty.kfupm.edu.sa › exams › Exam1_Sample1Discrete Math I – Practice Problems for Exam I - KFUPM

   faculty.kfupm.edu.sa › exams › Exam1_Sample1

   P(m,n) is the statement “m < n”. P(m,n) is the statement “m | n”. Suppose that the variable x represents students, F(x) means “x is a freshman,” and M(x) means “x is a math major”. For each of the three statements (a), (b), and (c), determine which of the symbolic statements are equivalent.

2. www.inf.ed.ac.uk › dmmr › slidesDiscrete Mathematics, Chapter 1.1.-1.3: Propositional Logic

   www.inf.ed.ac.uk › dmmr › slides

   If p and q are propositions, then p !q is a conditional statement or implication which is read as “if p, then q” and has this truth table: In p !q, p is the hypothesis (antecedent or premise) and q is the conclusion (or consequence). Implication can be expressed by disjunction and negation: p !q :p _q

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3. cs.rpi.edu › ~eanshel › 4020Sample Problems in Discrete Mathematics

   cs.rpi.edu › ~eanshel › 4020

   This handout lists some sample problems that you should be able to solve as a pre-requisite to Design and Analysis of Algorithms. Try to solve all of them. You should also read Chapters 2 and 3 of the textbook, and look at the Exercises at the end of these chapters.

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4. faculty.ksu.edu.sa › files › 31methods_of_proof151 Math Exercises (3,1) Methods of Proof

   faculty.ksu.edu.sa › files › 31methods_of_proof

   Direct Proof: A direct proof shows that a conditional statement p q is true by showing that if p is true, then q must also be true, so that the combination p true and q false never occurs. In a direct proof, we assume that p is true and use axioms, definitions, and previously proven theorems, together with rules of

5. circles.math.ucla.edu › data › Handout-2647-2178THE P VS. NP PROBLEM - University of California, Los Angeles

   circles.math.ucla.edu › data › Handout-2647-2178

   Nov 22, 2020 · P is the class of decision problems that can be solved with an algorithm that runs in polynomial time, in other words, O(nk) time for some constant k. Exercise 2. Show that the following decision problems are in P by designing an algo-rithm that solves them in polynomial time.

6. math.libretexts.org › Courses › Monroe_Community2.5: Logical Equivalences - Mathematics LibreTexts

   math.libretexts.org › Courses › Monroe_Community

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   hands-on exercise 2.5.3. The logical connective exclusive or, denoted p ⊻ q, means either p or q but not both. Consequently, p ⊻ q ≡ (p ∨ q) ∧ ¯ (p ∧ q) ≡ (p ∧ ¯ q) ∨ (¯ p ∧ q). Construct a truth table to verify this claim.

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   When is (P) false?

   • It is false only when \ (p\) is true and \ (q\) is false, and is true in all other situations. The statement \ (p\) in an implication \ (p \Rightarrow q\) is called its hypothesis, premise, or antecedent, and \ (q\) the conclusion or consequence. Implications come in many disguised forms.

   2.3: Implications - Mathematics LibreTexts

   math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/02:_Logic/2.03:_Implications

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   Is p q a logical equivalence?

   • Write p ⇒ q¯ ¯¯¯¯¯¯¯¯¯¯¯¯ as a conjunction. Two logical statements are logically equivalent if they always produce the same truth value. Consequently, p ≡ q is same as saying p ⇔ q is a tautology. Beside distributive and De Morgan’s laws, remember these two equivalences as well; they are very helpful when dealing with implications. Exercises 2.5.

   2.5: Logical Equivalences - Mathematics LibreTexts

   math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Discrete_Math/2:_Logic/2.5:_Logical_Equivalences

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   Why does a problem have to be in P vs NP?

   • Explain why if a problem is in P, then it must also be in NP. The P vs. NP conjecture simply says that the converse is false. That is, there are problems that can be checked in polynomial time, but cannot be solved directly in polynomial time. Exercise 6. Write an algorithm to solve the subset sum problem.

   THE P VS. NP PROBLEM

   circles.math.ucla.edu/circles/lib/data/Handout-2647-2178.pdf

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   What is P vs NP conjecture?

   • The P vs. NP conjecture simply says that the converse is false. That is, there are problems that can be checked in polynomial time, but cannot be solved directly in polynomial time. Exercise 6. Write an algorithm to solve the subset sum problem. Your algorithm should not run in polynomial time, unless you want to earn 1 million right now.

   THE P VS. NP PROBLEM

   circles.math.ucla.edu/circles/lib/data/Handout-2647-2178.pdf

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   What is logical equivalence of expand (p q) (r s)?

   • Expand (p ∨ q) ∧ (r ∨ s). is a tautology. We can use the properties of logical equivalence to show that this compound statement is logically equivalent to T. This kind of proof is usually more difficult to follow, so it is a good idea to supply the explanation in each step. Here is a complete proof:

   2.5: Logical Equivalences - Mathematics LibreTexts

   math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Discrete_Math/2:_Logic/2.5:_Logical_Equivalences

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   What is p q if and only if p $ q is a tautology?

   • p q if and only if p $ q is a tautology. Truth table proving De Morgan’s second law. The first of the Negation laws is also called “law of excluded middle”. Latin: “tertium non datur”. literal is either a propositional variable, or the negation of one. Examples: p, :p. clause is a disjunction of literals.

   Discrete Mathematics, Chapter 1.1.-1.3: Propositional Logic

   www.inf.ed.ac.uk/teaching/courses/dmmr/slides/13-14/Ch1a.pdf

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8. math.libretexts.org › Bookshelves › Combinatorics_and2.3: Implications - Mathematics LibreTexts

   math.libretexts.org › Bookshelves › Combinatorics_and

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   Conditional statements are also called implications. An implication is the compound statement of the form “if p, then q.”. It is denoted p ⇒ q, which is read as “ p implies q.”. It is false only when p is true and q is false, and is true in all other situations. p p.