Chapter 2 - The Logic of Compound Statements

What is the associative laws?

(p∧q)∧r ≡ p∧(q∧r) and (p∨q)∨r ≡ p∨(q∨r)

True or false? The negation of: "If Sue is Luiz's mother, then Ali is his cousin" is "If Sue is Luiz's mother, then Ali is not his cousin."

False. The negation of an if-then statement is not an if-then statement. It is an and statement.

For an argument to be invalid means that there is an argument of the same form whose premises ______ and whose conclusion __________ .

For an argument to be invalid means that there is an argument of the same form whose premises are all true and whose conclusion is false.

For an argument to be invalid means that there is an argument of the same form whose premises _______ and whose conclusion ______ .

For an argument to be invalid means that there is an argument of the same form whose premises are all true and whose conclusion is false.

For an argument to be sound means that it is _______ and its premises ______. . In this case we can be sure that its conclusion ______ .

For an argument to be sound means that it is valid and its premises are all true. In this case we can be sure that is conclusion is true.

For an argument to be sound means that it is _______ and its premises ________. In this case we can be sure that its conclusion _________.

For an argument to be sound means that it is valid and its premises are all true. In this case we can be sure that its conclusion is true.

For an argument to be valid means that every argument of the same form whose premises ______ has a conclusion ____.

For an argument to be valid means that every argument of the same form whose premises are all true has a conclusion that is true.

For an argument to be valid means that every argument of the same form whose premises ______ has a ______ conclusion.

For an argument to be valid means that every argument of the same form whose premises are all true has a true conclusion.

Use De Morgan's laws to write negations for the statement: Hal is a math major and Hal's sister is a computer science major.

Hal is not a math major or Hal's sister is not a computer science major.

Rewrite the following sentence in if-then form: Catching the 8:05 bus is a sufficient condition for my being on time for work.

If I catch the 8:05 bus then I am on time for work.

Rewrite the following sentence in if-then form: A sufficient condition for Jon's team to win the championship is that it win the rest of its games.

If Jon's team win the rest of its games then they win the championship.

"R is a sufficient condition for S" means

If R then S

"R only if S" means

If R then S

"R is a necessary condition for S" means

If S then R

Rewrite the following sentence in if-then form: Payment will be made on the fifth unless a new hearing is granted.

If a new hearing is not granted, payment will be made on the fifth.

Write the contrapositive of the following sentence: If Tom is Ann's father, then Jim is her uncle and Sue is her aunt.

If either Jim is not Ann's uncle or Sue not her aunt, then Tom is not her father.

Write the contrapositive of the following sentence: If n is prime, then n is odd or n is 2.

If n is not odd and n is not 2 then n is not prime.

What is a conjunction?

If p and q are statement variables, the conjunction of p and q is "p and q," denoted p ∧ q. It is true when, and only when, both p and q are true. If either p or q is false, or if both are false, p ∧ q is false.

What is a disjunction?

If p and q are statement variables, the disjunction of p and q is "p or q," denoted p ∨ q. It is true when either p is true, or q is true, or both p and q are true; it is false only when both p and q are false.

Write the contrapositive of the following sentence: If P is a square, then P is a rectangle.

If p is not a rectangle, then P is not a square.

Rewrite the following in argument form using letters: If all integers are rational, then the number 1 is rational. All integers are rational. Therefore, the number 1 is rational.

If p then q. p. Therefore, q.

Rewrite the statement in if-then form in two ways, one of which is the contrapositive of the other. The Cubs will win the pennant only if they win tomorrow's game.

If the Cubs do not win tomorrow's game, then they will not win the pennant. If the Cub's win the pennant, then they will have won tomorrow's game.

Rewrite the following statement in if-then form: This loop will repeat exactly N times if it does not contain a stop or a go to.

If this loop does not contain a stop or a go to then it will repeat exactly N times.

Use the contrapositive to rewrite the statements in if-then form in two ways. Being divisible by 3 is a necessary condition for this number to be divisible by 9.

If this number is not divisible by 3, then it is not divisible by 9. If this number is divisible by 9, then it is divisible by 3.

Rewrite the statement as a conjunction of two if-then statements. This quadratic equation has two distinct real roots if, and only if, its discriminant is greater than zero.

If this quadratic equation has two distinct real roots, then its discriminant is greater than zero, and if the discriminant of this quadratic equation is greater than zero, then the equation has two real roots.

Rewrite the following statement in if-then form: Freeze or I'll shoot.

If you don't freeze then I will shoot.

When are two statements logically equivalent?

If, and only if, they have identical truth values for each possible substitution of statements for their statement variables. The logical equivalence of statement forms P and Q is denoted by writing P ≡ Q.

In the following sentence, is the word or used in its inclusive or exclusive sense? A team wins the playoffs if it wins two games in a row or a total of three games.

Inclusive or. For instance, a team could win the playoff by winning games 1, 3 and 4 and losing game 2. Such an outcome would satisfy both conditions.

If logic is easy, then I am a monkey's uncle. I am not a monkey's uncle. ∴_______________

Logic is not easy

A conditional statement and its contrapositive are

Logically equivalent

A conditional statement and its converse are not

Logically equivalent

Write a negation for the following sentence: If P is a square, then P is a rectangle.

P is a square and P is not a rectangle.

Indicate if the following sentences are statements: 1,024 is the smallest four-digit number that is a perfect square.

This is a statement because it is a true sentence.

Write a negation for the following sentence: If Tom is Anna's father, then Jim is her uncle and Sue is her aunt.

Tom is Anna's father and either Jim is not her uncle or Sue is not her aunt.

Rewrite the following sentence using variables and verify is validly: If Tom is not on team A, then Hua is on team B. If Hua is not on team B, then Tom is on team A. ∴ Tom is not on team A or Hua is not on team B.

~p → q ~q → p ∴ ~p v ~q

What is the inverse of p -> q?

~p → ~q

What is the contrapositive of p -> q

~q -> ~p

Use De Morgan's laws to write negations for the statement: The connector is loose or the machine is unplugged.

The connector is not loose and the machine is not unplugged in.

What does neither p nor q equal to?

~p ^ ~q

Rewrite the following sentence using variables and decide the logic error: If this number is larger than 2, then its square is larger than 4. This number is not larger than 2. ∴ The square of this number is not larger than 4.

p → q ~p ∴ ~q Invalid: Inverse error

What is the commutative laws?

p ∧ q ≡ q ∧ p and p ∨ q ≡ q ∨ p

What is the negation of p -> q ?

p ∧ ∼q

Give example of Modus Ponens in argument form:

p→q p ∴q

Give example of Modus Tollens in argument form:

p→q ∼q ∴ ∼p

The negation of "if p then q" is?

p∧ ∼q

What is the converse of p -> q?

q -> p

What is a statement or proposition?

A sentence that is true or false but not both.

What is a contradiction?

A statement form that is always false regardless of the truth values of the individual statements substituted for its statement variables.

Let h = "John is healthy," w = "John is wealthy," and s = "John is wise." A. John is healthy and wealthy but not wise. B. John is neither wealthy nor wise, but he is healthy.

A. (h^w) ^ ~s B. (~w ^ ~s) ^ h

Let p be the statement "DATAENDFLAG is off," q the statement "ERROR equals 0," and r the statement "SUM is less than 1,000." Express the following sentences in symbolic notation: A. DATAENDFLAG is off, ERROR equals 0, and SUM is less than 1,000. C. DATAENDFLAG is off; however, ERROR is not 0 or SUM is greater than or equal to 1,000.

A. p ^ q ^ r B. p ^ (~q v ~r)

Rewrite the following in argument form using letters: A. This number is even or this number is odd. This number is not even. Therefore, this number is odd. B. ______ or logic is confusing My mind is not shot. Therefore, ____________

A: p v q ~p q B: My mind is shot or logic is confusing. My mind is not shot. Therefore, logic is confusing

Write the statements in symbolic form using the symbols ∼, ∨, and ∧ and the indicated letters to represent component statements. Let s = "stocks are increasing" and i = "interest rates are steady." A. Stocks are increasing but interest rates are steady. B. Neither are stocks increasing nor are interest rates steady.

A: s ^ i B: ~s ^ ~i

Suppose that p and q are statements so that p → q is false. Find the truth table for ∼p → q

Because p → q is false, p is true and q is false. Hence ~p is false and so ~p → q is true.

Write the converse and inverse of the following sentence: If Tom is Ann's father, then Jim is her uncle and Sue is her aunt.

Converse: If Jim is Anna's uncle and Sue is her aunt then Tom is Ann's father. Inverse: If Tom isn't Ann's father then Jim is not her uncle or Sue is her aunt.

Write the converse and inverse of the following sentence: If P is a square, then P is a rectangle.

Converse: If P is a rectangle then P is a square. Inverse: If P is not a square then P is not a rectangle.

Write the converse and inverse of the following sentence: If n is prime, then n is odd or n is 2.

Converse: If n is odd or n is 2 then n is prime. Inverse: If n is not prime then n is not odd and n is not 2

The converse of "if p then q" is?

if q then p

The inverse of "if p then q" is?

if ∼p then ∼q

The contrapositive of "if p then q" is?

if ∼q then ∼p

Write a negation for the following sentence: If n is prime, then n is odd or n is 2.

n is prime and both n is not odd and n is not 2 or n is prime and n is neither odd nor 2

Write the following statements in symbolic form and decide if they are logically equivalent: If you paid full price, you didn't buy it at Crown Books. You didn't buy it at Crown Books or you paid full price.

p -> q q v p

What is p but q equal to?

p ^ q

Rewrite the following sentence using variables and decide if the statement is valid or invalid: This real number is rational or it is irrational. This real number is not rational. ∴ This real number is irrational.

p v q ~p ∴ q Valid: Elimination

Rewrite the following sentence using variables and decide if statement is valid or invalid: If Jules solved this problem correctly, then Jules obtained the answer 2. Jules obtained the answer 2. ∴ Jules solved this problem correctly.

p → q q ∴ p Invalid: converse error

Rewrite the following sentence using variables and decide if the statement is valid or invalid: If I go to the movies, I won't finish my homework. If I don't finish my homework, I won't do well on the exam tomorrow. ∴ If I go to the movies, I won't do well on the exam tomorrow.

p → q q → r ∴ q → r Valid: Transitivity

What is De Morgan's Laws?

∼( p ∧ q ) ≡ ∼ p ∨ ∼q and ∼( p ∨ q ) ≡ ∼ p ∧ ∼q

What is De Morgan's Laws?

∼( p ∧ q ) ≡ ∼ p ∨ ∼q and ∼( p ∨ q ) ≡ ∼ p ∧ ∼q