[PDF] Logical Inference and Mathematical Proof Need for inference

Solution of Assignment #2 CS/191

by the implication law (the first law in Table 7.) ≡q ∨ (¬p) by commutative Show that (p ∨ q) ∧ (¬p ∨ r) → (q ∨ r) is a tautology. sol: (p ∨ q) ...

2. Propositional Equivalences 2.1. Tautology/Contradiction

5. Prove [(p → q) ∧ (q → r)] ⇒ (p → r) using a truth table. Exercise 2.6.6.

Propositional Logic Discrete Mathematics

Prove that: [(p → q) ∧ (q → r)] → [p → r] is a tautology. By using truth table. By using logic equivalence laws. We will show these examples in class. c 

Chapter 1 Logic

Similarly (q ∨ r) ∧ p ⇔ (q ∧ p) ∨ (r ∧ p). The Laws of Logic can be used in several other ways. One of them is to prove that a statement is a tautology 

Math 55: Discrete Mathematics

By the definition of conditional statements on page 6 using the Com- mutativity Law

Basic Argument Forms

if p then q; and if r then s; but either not q or not s; therefore either not p or not r. Simplification. (p ∧ q). ∴ p p and q are true; therefore p is 

7. Let p and q be the propositions - p: It is below freezing. q

(p^q) ^r = p ^ (q ^ r) EXAMPLE 6*. Show that (p ^ q) → (p ≤ q) is a tautology. Solution: To show that this statement is a tautology we will use logical ...

MA0301 ELEMENTARY DISCRETE MATHEMATICS NTNU

Jan 6 2020 ≡ ¬p ∨ q. (Using that (¬p ∨ p) is a tautology). D. Exercise 9. Use the laws of logic to simplify (s ∨ (p ∧ r ∧ s)) ∧ (p ∨ (p ∧ q ∧ ¬r) ...

Inference Rules and Proof Methods

The argument is valid since ((p → q) ∧ p) → q is a tautology. CSI2101 A real number r is rational if there exists integers p and q with q = 0 such.

Solution of Assignment #2 CS/191

Since [(p ? q) ? (q ? r)] ? (p ? r) is always T it is a tautology. (0 points) (c) by the implication law (the first law in Table 7.) ?q ? (¬p).

CSE 311: Foundations of Computing I Section: Gates and

q ? (p ? r) following propositional formulae are tautologies by showing they are equivalent ... simplify it using axioms and laws of boolean algebra.

Propositional Logic Discrete Mathematics

Prove that: [(p ? q) ? (q ? r)] ? [p ? r] is a tautology. By using truth table. By using logic equivalence laws. We will show these examples in class. c 

2. Propositional Equivalences 2.1. Tautology/Contradiction

Example 2.1.2. p ? ¬p. Definition 2.1.3. A contingency is a proposition that is neither a tautology nor a contradiction. Example 2.1.3. p ? q ? ¬r.

(1) Propositional Logic

) Show that ( p ? q ) ? ( p ? r ) and p ? ( q ? r ) are logically equivalent ? Solution: Page 22. Math 151 Discrete Mathematics ( Propositional Logic ). By 

Midterm Exam

(4 points) Show that (P ? (Q ? R)) ? ((P ?Q) ? R) is tautology using logical (4 points) Validate the following argument by rules of inference ...

Lecture 5 - 188 200 Discrete Mathematics and Linear Algebra

The rules of logic specify the meaning of mathematical statement. (equivalent). Example p ? q. ? r. ? p ? (q ? r). Pattarawit Polpinit. Lecture 5 ...

Math 55: Discrete Mathematics

d) q ? p: If the votes are counted then the election is decided. e) ¬q ? ¬p: The 1.3.30 Show that (p ? q) ? (¬p ? r) ? (q ? r) is a tautology.

Chapter 1 Logic

p ? ¬q. Using the same reasoning or by negating the negation

MATH 363 Discrete Mathematics SOLUTIONS: Assingment 1 1

(2pt each) Write these propositions using r s

Methods of proof - Michigan State University

Prove: If p ?r and q ?¬r then p ?q ?s Equivalently prove: (p ?r) ?(q ?¬r ) ?(p ?q ?s) 1 p ?r Premise 2 ¬p ?r 1 Implication 3 q ?¬r Premise 4 ¬q ?¬r 3 Implication 5 ¬p ?¬q 2 4 Resolution 6 ¬(p ?q ) 5 DeMorgan

P Q R)) P Q R - University of Oxford

p q r q p r ? q aka Disjunction Elimination Corresponding Tautology: ((p q) ? (r q) ? (p r )) q Example: Let p be “I will study discrete math ” Let q be “I will study Computer Science ” Let r be “I will study databases ” “If I will study discrete math then I will study Computer Science ”

13 Propositional Equivalences - University of Hawai?i

Tautologies Contradictions and Contingencies A tautology is a compound proposition which is always true A contradiction is a compound proposition which is always false A contingency is a compound proposition which is neither a tautology nor a contradiction Logical Equivalences

2 Propositional Equivalences 21 Tautology/Contradiction

Example 2 1 3 p_q!:r Discussion One of the important techniques used in proving theorems is to replace or sub-stitute one proposition by another one that is equivalent to it In this section we will list some of the basic propositional equivalences and show how they can be used to prove other equivalences

P Q R)) P Q R - University of Oxford

(P ? (Q ? R)) ? (P ?Q ? R) is a tautology A sentence of the language of propositional logic is a tautology (logically true) if and only if the main column has T in every line of the truth value (that is if and only if the sentence is true in any L Ô-structure)

Solutions to problem set 2 Problem 1 Equivalences

tation they lead to the same value Hint: use truth table to show the equivalence P R Q (P ? R) Q ? R (P ? R)? Q ? R (P ? Q) ? R) We can prove

What is a tautology in propositional logic?

A sentence of the language of propositional logic is a tautology (logically true) if and only if the main column has T in every line of the truth value (that is, if and only if the sentence is true in any L. Ô-structure). Ø(P ?(Q ?R)) ?(P ? Q ?R) As it stands, the sentence (P ? (Q ? R)) ? (P ?Q ? R) is merely in abbreviated form.

What is the difference between a tautology and a contradiction?

1.3 Propositional Equivalences Tautologies, Contradictions, and Contingencies A tautology is a compound proposition which is always true. A contradiction is a compound proposition which is always false. A contingency is a compound proposition which is neither a tautology nor a contradiction.

What does tautology mean?

?a tautology, or ?an axiom/law of the domain (e.g., 1+3=4 orx> +1 ) ?justified by definition, or ?logically equivalent to orimpliedby one or more propositions  pk

What is the best way to prove P?R?

?Prove: If  p?rand ¬r, then q?¬p ?Equivalently, prove: (p?r) ?( ¬r ) ?( q?¬p) 1. p?r Premise 2. ¬r Premise 3. ¬p1, 2, modus tollens