Elementary Logic
... what we said above about the converse, it IS true that a statement and its contrapositive are equivalent. One way to see this is to notice that A ⇒ B is false only if A is true and B is false (see above), and the contrapositive (Not B) ⇒ (Not A) is false only if (Not B) is true and (Not A) is false ...
... what we said above about the converse, it IS true that a statement and its contrapositive are equivalent. One way to see this is to notice that A ⇒ B is false only if A is true and B is false (see above), and the contrapositive (Not B) ⇒ (Not A) is false only if (Not B) is true and (Not A) is false ...
MODULE I
... 1) Obtain the conjunctive normal form of P (P→ Q) P (┐PQ) 2) QP┐Q) ┐P┐Q)) 3) ┐( PQ) PQ Minterms minterm consists of conjunction in which each statement variable or its negation , not both appears only once. For two variables P and Q there are 22 minterms. These are known ...
... 1) Obtain the conjunctive normal form of P (P→ Q) P (┐PQ) 2) QP┐Q) ┐P┐Q)) 3) ┐( PQ) PQ Minterms minterm consists of conjunction in which each statement variable or its negation , not both appears only once. For two variables P and Q there are 22 minterms. These are known ...
3.1.3 Subformulas
... speaking, a subformula of a propositional formula H is a string occurring in H which itself is a propositional formula. Definition 3.8 Let F be a propositional formula. The set of subformulas of F is the smallest set S(F ) satisfying the following conditions: 1. F ∈ S(F ). 2. If ¬G ∈ S(F ) , then G ...
... speaking, a subformula of a propositional formula H is a string occurring in H which itself is a propositional formula. Definition 3.8 Let F be a propositional formula. The set of subformulas of F is the smallest set S(F ) satisfying the following conditions: 1. F ∈ S(F ). 2. If ¬G ∈ S(F ) , then G ...
The Closed World Assumption
... We view our program as a logical theory expressing knowledge about the world. In several situations, it is convenient to assume that the program contains complete information about certain kinds of logical statements. We can then make additional inferences about the world based on the assumed comple ...
... We view our program as a logical theory expressing knowledge about the world. In several situations, it is convenient to assume that the program contains complete information about certain kinds of logical statements. We can then make additional inferences about the world based on the assumed comple ...
Supplement: Conditional statements and basic methods of proof
... First note that there is only one set of circumstances under which a conditional statement is false: The hypothesis is true and the conclusion is false. Therefore to establish that a conditional statement is false, it suffices to produce a specific counterexample; that is, a specific situation for w ...
... First note that there is only one set of circumstances under which a conditional statement is false: The hypothesis is true and the conclusion is false. Therefore to establish that a conditional statement is false, it suffices to produce a specific counterexample; that is, a specific situation for w ...
Automated Reasoning Lecture 5: First
... the property of being even to P i.e. P I ≡ {0, 2, 4, . . .}, and the relation of being greater than to Q, i.e. the set of pairs Q I ≡ {(1, 0), . . . , (2, 0), (2, 1), . . . , (89, 27), . . . }; ...
... the property of being even to P i.e. P I ≡ {0, 2, 4, . . .}, and the relation of being greater than to Q, i.e. the set of pairs Q I ≡ {(1, 0), . . . , (2, 0), (2, 1), . . . , (89, 27), . . . }; ...
8.5 Applications of Systems of Linear Equations
... For the 2009 Major League Baseball and National Football League seasons, based on average ticket prices, three baseball tickets and two football tickets would have cost $229.90. Two baseball tickets and one football ticket would have cost $128.27. What were the average ticket prices for the tickets ...
... For the 2009 Major League Baseball and National Football League seasons, based on average ticket prices, three baseball tickets and two football tickets would have cost $229.90. Two baseball tickets and one football ticket would have cost $128.27. What were the average ticket prices for the tickets ...
An Instantiation-Based Theorem Prover for First
... These limitations in MILP can be addressed by switching to a more expressive language like first-order logic (FOL). FOL lets us specify a problem in terms of classes and relations, and reason about these classes and relations directly—we can do lifted reasoning. Lifted reasoning lets us work with st ...
... These limitations in MILP can be addressed by switching to a more expressive language like first-order logic (FOL). FOL lets us specify a problem in terms of classes and relations, and reason about these classes and relations directly—we can do lifted reasoning. Lifted reasoning lets us work with st ...
Chapter 4, Propositional Calculus
... 2.1. Proposition (or statement) = a declarative statement (in contrast to a command, a question, or an exclamation) which is true or false, but not both. 2.1.1. Examples: “Obama is president.” is a proposition. “Obama will be re-elected.” is not a proposition. 2.1.2. “This statement is false.” is a ...
... 2.1. Proposition (or statement) = a declarative statement (in contrast to a command, a question, or an exclamation) which is true or false, but not both. 2.1.1. Examples: “Obama is president.” is a proposition. “Obama will be re-elected.” is not a proposition. 2.1.2. “This statement is false.” is a ...
Mathematics for Computer Science/Software Engineering
... statement ‘if p is true then q is true’ obviously cannot be true, and therefore must be false. On the other hand, if p is false, then the statement ‘if p is true then ...’ is an empty statement—it is saying nothing at all, and therefore cannot be false. So it must be true. If you work out the truth ...
... statement ‘if p is true then q is true’ obviously cannot be true, and therefore must be false. On the other hand, if p is false, then the statement ‘if p is true then ...’ is an empty statement—it is saying nothing at all, and therefore cannot be false. So it must be true. If you work out the truth ...
