If P Then Q If Q Then R Therefore If P Then R Proof

Answer (1 of 6) The easiest way is by recalling that the contrapositive has the same truth value as the original statement So since P implies Q is true then not Q implies not P is also true However, the question remains how do we know that the contrapositive is true given the statement is tru.

If p then q if q then r therefore if p then r proof. If an independent premise is removed, the support that other premises supply to the conclusion is not affected.  If ( P implies Q ) then ( P implies R ) You cannot infer that If Q then R Again this is because the first is always true when P is false, but choosing Q and R appropriately makes the second false Also, you are very confused about logic. In an implication p ® q , if we can establish that q is true , then regardless of the truth value of p, the implication p ® q So the construction of a trivial proof of p ® q needs to show that the truth value of q is true Vacous proof If the hypothesis p of an implication p ® q is false , then p ® q is true for any proposition q.

1) q, if p 2) q, provided p 3) p, only of q 4) p implies q 5) q whenever, p 6. Q → R ––––––––– ~Q Proof Suppose that the premises are all true Then ~R and Q →R are both true, but then ~Q is true, in virtue of modus tollens Thus, if the premises are all true, then the conclusion is also true, which is to say the argument is valid. If q then r Therefore if p then r Example ¾ 1 If the Conservatives won the from SSH 105 at Ryerson University.

P, therefore q The conditional statemen, p → q also means;. Truth Table Method p q r p →q q →r p r. P → q (p implies q) (if p then q) is the proposition that is false when p is true and q is false and true otherwise Equivalent to finot p or qfl Ex If I am elected then I will lower the taxes If you get 100% on the final then you will get an A p I am elected q I will lower the taxes Think of it as a contract, obligation or pledge.

Solution Note that while we could start rewriting these statements with logically equivalent replacements in the hopes of transforming one into another, we will never be sure that our failure is due to their lack of logical equivalence rather than our lack of imagination. R When in doubt, use parenthesis c Xin He (University at Buffalo) CSE 191 Discrete Structures 19 / 37 Translating logical formulas to English sentences Using the above logic operators, we can construct more complicated logical formulas (They are calledcompound propositions) Example Proposition p Alice is smart. If we call the first part p and the second part q then we know that p results in q This means that if p is true then q will also be true This is called the law of detachment and is noted $$\left (p \to q)\wedge p \right \to q$$ The law of syllogism tells us that if p → q and q → r then p → r is also true This is noted.

Otherwise it is true. R means (p ^ q ) !. Occurs when one or more steps of a proof are based on the truth of the statement being proved Methods of Proof of an Implication Generally, we will be solving problems of the form, ``If p, then q'' where p and q are statements Symbolically we write such implications as p > q where > is called the implication operator.

As far as I understand, If p then Q means " if P is true, Q has to be true Any other case, I don't know " So, from what I understand, the first 2 rows of the truth table state that " If P is true and Q is true, the outcome is correct and If P is true and Q is false,. P ^ q !. Proof by Cases 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”.

How to think about "P ⊃ Q" in plain EnglishIn propositional logic, P ⊃ Q is what is called a material implicationIt doesn't mean that P and Q mean the same thing (they might not have the same truth value);. Sec 36 Analyzing Arguments with Truth Tables Some arguments are more easily analyzed to determine if they are valid or invalid using Truth Tables instead of Euler Diagrams One example of such an argument is. If p (antecedent) and q (consequent) be proposition variables (or sentential variables), the conditional of q by p is "If p then q" or "p implies q" We symbolize the conditional by p → q, and frequently read "p implies q" or It is false when p is true and q is false;.

 p, therefore q q, if p p is enough condition for q Reason for the above selection The conditional statement p → q is read as If p then q The deductive validity of the conditional statement is;. コンプリート！ if p then q if q then r therefore if p then r If p then q or r. If p then q p Therefore q eg All humans are mortal Socrates is a human Therefore, Socrates is mortal 2 Modus Tollens (“Method of denying”) e ∼p ∧ r →∼s f ∴ ∼q 4) Formal Proof • A formal proof of a conclusion C, given premises p1, p2,,pn consists of a sequence of steps, each of.

P→Q means If P then Q ~R means NotR P ∧ Q means P and Q P ∨ Q means P or Q An argument is valid if the following conditional holds If all the premises are true, the conclusion must be true Some valid argument forms (1) 1 P 2 P→Q C Therefore, Q. If p then q is equiv to not p or q Exportation ((p∧q) → r) ∴ (p → (q → r)) from (if p and q are true then r is true) we can prove (if q is true then r is true, if p is true) Importation (p → (q → r)) ∴ ((p∧q) → r) Tautology (Ô) p ∴ (p∨p) p is true is equiv to p is true or p is true Tautology (ò) p ∴ (p∧p). Direct proof of P )Q This is the simplest, and most natural, method of proof Try this rst before considering other methods Assume P Logical deductions Therefore Q Proof by contraposition of P )Q The contraposition (or contrapositive) of an implication (1) P )Q is the implication (2) Q )P (where denotes negation).

1 q e 2 2 2q er r) 2 = q2e 1 1 q 2e 2 2 q er r Therefore every prime in the prime factorization of n is raised to an even exponent Conversely suppose that every prime in the prime factorization of n is raised to an even exponent Then n = p2k 1 1 p 2k 2 2 p 2kr r where p i are primes and k j are positive integers Let m = pk 1 1 p k 2 2 p kr. Otherwise it is true The contrapositive of a conditional statement of the form "If p then q ". The last statement is the conclusion and all its preceding statements are called premises (or hypothesis) The symbol “ ∴ ”, (read therefore) is placed before the conclusion A valid argument is one where the conclusion follows from the truth values of the premises Rules of Inference provide the templates or guidelines for constructing.

 If p or q, then r Therefore, if not r , then not p and not q Example 211 Identifying Logical Form Fill in the blanks below so that argument (b) has the same form as argument (a) Then represent the common form of the arguments using letters to stand for component sentences a. P ∨q ¬p∨r∴q∨r is called resolution and is used in computer (automatic) theorem proving/reasoning • also basis of logical programming languages like Prolog If all hypotheses and the conclusion are expressed as clauses (disjunction of variables or their negations), we can use resolution as the only rule of inference. Ment which asserts that if pis true, then q is also true We agree that p!qis true when pis false The statement pis called the hypothesis of the implication, Therefore (p_q) is the same as 4 CHAPTER 1 LOGIC p^q Using the same reasoning, or by negating the negation, we can see.

Assume √2is rational Then there exists integers p and q such that q ≠ 0, p / q = √ , and p and q have no common divisors other than 1 and 1 Since p / q = √2 and q ≠ 0, we have p = √2q, so p2 = 2q2 Since q2 is an integer and p2 = 2q2, we have that p2 is even By our earlier result, since p2 is. In instances of modus tollens we assume as premises that p → q is true and q is false There is only one line of the truth table—the fourth line—which satisfies these two conditions In this line, p is false Therefore, in every instance in which p → q is true and q is false, p must also be false Formal proof Via disjunctive syllogism. The sentence "If (if P, then Q) and (if Q, then R), then (if P, then R)" captures the principle of the previous paragraph It is an example of a tautology, a sentence which is always true regardless of the truth of P, Q, and R Here is a table that establishes this tautology "IFTHEN" Is Transitive P.

All that it is, is a claim that if P is true, then Q is also true — without making any more claims than this An alternative way of considering P ⊃ Q is as a "constraint" that. If p, then q If q, then r Therefore, if p, then r A deductively valid argument with true premises is said to be. View Homework Help HW_12pdf from PHI 10 at Miami Dade College, Miami Therefore, not p Hypothetical Syllogism If p, then q If q, then r Therefore, if p, then r Disjunctive Syllogism Either.

Valid Form Invalid Cousin Modus Ponens If P, then Q P _____ Q Affirming the Consequent If P, then Q Q _____ P Modus Tollens If P, then Q. Let p and q be statement variables which apply to the following definitions The conditional of q by p is "If p then q " or " p implies q " and is denoted by p q It is false when p is true and q is false;. A formal proof of a conclusion q given hypotheses p 1;p 2;;p n is a sequence of steps, each of which applies some inference rule to hypotheses or previously proven statements (antecedents) to yield a new true statement (the consequent) A formal proof demonstrates that if the premises are true, then the conclusion is true.

 Result 26 (Transitivity) Suppose p, q and r are statement forms Then the following argument (called transitivity) is valid p → q q → r p → r Result 27 (Proof by Division into Cases) Suppose p, q and r are statement forms Then the following argument (called proofby division into cases) is valid p∨q p → r q → r r Result 28In this case, \((P \imp R) \vee (Q \imp R)\) is true, but. If p then q If q then r Therefore, if p then r independent premise A premise that does not depend in other premises to provide support to a conclusion If this premise is removed the support that other premies supply to the conclusion is not affected inductive argument. A valid argument form made up of three hypothetical, or conditional, statements If p, then q If q, then r Therefore, if p, then r Independent Premise A premise that does not depend on other premises to provide support to a conclusion;.

Type 1 Modus Ponens If p, then q p Therefore, q Type 2 Modus Tollens If p, then q Not q Therefore, not p Type 3 Hypothetical Argument If. If p or q, then r Therefore, if not r, then not p and not q Practice Exercises D E F I N I T I O N S Argument a sequence of statements aimed at demonstrating the truth of a statement or assertion Statement a sentence that is either true or false, but not both It is also called a. A Famous and Beautiful Proof Theorem √2 is irrational Proof By contradiction;.

P then q” or “p implies q”, represented “p → q” is called a conditional proposition For instance “if John is from Chicago then John is from Illinois” The proposition p is called hypothesis or antecedent, and the proposition q is the conclusion or consequent Note that p → q is true always except when p is true and q is false. " If not q and p implies q, then not p Example p = it is sunny, q = it is hot p → q, it is hot whenever it is sunny “Given the above, if it is not hot, it cannot be sunny” 11 Hypothe’calSyllogism " If p implies q, and q implies r, then p implies r Example p = it is sunny, q = it is hot, r = it is dry. Given p →q, q →r, p Prove r We want to establish the logical implication (p →q)∧(q →r)∧p ⇒r We can use either of the following approaches Truth Table A chain of logical implications Note that if A⇒B andB⇒C then A⇒C MSU/CSE 260 Fall 09 10 Does (p →q)∧(q →r)∧p⇒r ?.

Result 26 (Transitivity) Suppose p, q and r are statement forms Then the following argument (called transitivity) is valid p → q q → r p → r Result 27 (Proof by Division into Cases) Suppose p, q and r are statement forms Then the following argument (called proofby division into cases) is valid p∨q p → r q → r r Result 28. And if p then r;. ) p Transitivity p !q Proof by Division into Cases p_q q !r p !q ) p !r q !r Conjunction p) r q Contradiction Rule ˘p !c ) p^q) p 23 In Class Group Work The famous detective Percule Hoirot was called in to solve a ba ing murder mystery He determined the following facts (1) Lord Hazelton, the murdered man, was killed by a blow on the head.

If p then q;. Therefore if p is true then q and r are true we can prove (if q is true then r is true, if p is true) Importation p → (q → r) ├ (p ∧ q) → r Tautology p ├ (p ∨ p) p is true is equiv to p is true or p is true Axiomatization. Are the statements \((P \vee Q) \imp R\) and \((P \imp R) \vee (Q \imp R)\) logically equivalent?.