MATH 3050 Discrete Mathematics Section 101 ( Barsamian )

Class Discussion Topics

In MATH 3050 Section 101, the class meetings will be run as seminars, not lectures, so your participation is essential. I will give each of you a small assignment to prepare for discussion in each class meeting. Starting with Day 2 ( Wednesday, January 11 ), you will have a class participation score for each class meeting, a score that indicates your contribution to the classroom discussion. The class participation score for each class meeting will be either 0, 1, 2, or 3, computed as follows:

- 0 point: Did not attend class.
- 1 point: Attended class but did not participate in discussion.
- 2 points: Attended class and participated in discussion, but had not prepared for discussion.
- 3 points: Attended class and participated in discussion, with adequate preparation.

The daily assignments are listed below.

Section 2.2 Conditional Statements

**Bowsher:**Consider these two statements:- A: If the moon is made of green cheese, then the Chicago Cubs win the 2016 World Series.
- B: If the moon is made of green cheese, then the world is flat.

**Carpenter:**There is a special term for the situation where a conditional statement that is true by virtue of the fact that its hypothesis is false. What is the term ? ( Reference the book in your explanation. )**Cicigoi:**We have encountered four logical operators so far: ∧, ∨, →, ~. What is the order of operations ? ( Reference the book in your explanation. ) What is the order of operations for the statement form*p*∧ ~*q*→*r*? ( Use parentheses to clarify the order of operations. )**Dewitt:**Construct a truth table for*p*∧ ~*q*→*r***Geist:**Use a truth table to prove that*p*→*q*is logically equivalent to ~*p*∨*q***Grueser:**Given that*p*→*q*≡ ~*p*∨*q*, use DeMorgan's Laws to f∈*D*~ (*p*→*q*)**Hisey:**Negate this logical statment: If Bob is green then Carol is red.- Given Conditional Statement
*S*:*p*→*q*,**Klions:**Make a truth table for*S***Korzan:**Write the*Converse of S*and make a truth table for it.**Silveira:**Write the*Contrapositive of S*and make a truth table for it.**Strobel:**Write the*Inverse of S*and make a truth table for it.**Weldon:**Write the*Negation of S*and make a truth table for it.

**Wohl:**Su*p*ose that the conditional statement*p*→*q*is false. What are values of*p*,*q*?**Worthington:**Su*p*ose that the conditional statement*p*→*q*is true. Can you determine the values of*p*,*q*? Explain.

Section 2.3 Valid and Invalid Arguments

**One of you will answer this general question about argument forms:**

**Carpenter:**How does one test an argument form to determine if it is valid ?

**Eight of you will answer questions about these four sample argument forms:**

Example 1 | Example 2 | Example 3 | Example 4 |
---|---|---|---|

p → qp∴ q |
p → q~q∴ ~p |
p → qq∴ p |
p → q~p∴ ~q |

**Cicigoi:**Make a truth table to determine if Example #1 is a valid argument form.**Dewitt:**Make a truth table to determine if Example #2 is a valid argument form.**Geist:**Make a truth table to determine if Example #3 is a valid argument form.**Grueser:**Make a truth table to determine if Example #4 is a valid argument form.**Hisey:**Valid or not, the argument form in Example #1 is famous. What is it called ?**Klions:**Valid or not, the argument form in Example #2 is famous. What is it called ?**Korzan:**Valid or not, the argument form in Example #3 is famous. What is it called ?**Silveira:**Valid or not, the argument form in Example #4 is famous. What is it called ?

**Four of you will answer questions about valid arguments.**

**Strobel:**What is a*valid argument*? How is it different from a*valid argument form*?**Weldon:**Give an example of an invalid argument that commits the Converse Error. ( Make up your own example. )**Wohl:**Give an example of an invalid argument that commits the Inverse Error. ( Make up your own example. )**Worthington:**Give an example of a valid argument that has false premises and a false conclusion. ( Make up your own example. )**Bowsher:**Give an example of an invalid argument that has true premises and a true conclusion. ( Make up your own example. )

Section 3.1 Predicates and Quantified Statements I

**Cicigoi:**Make up two examples of predicates, as follows:- Make up an example of a predicate that involves a real variable ( a variable that represents a real number ).
- Make up a different example of a predicate that involves a variable that represents something else, not a number.

**Dewitt:**Make up an example of a predicate involving a simple statement ( not a conditional ) with a real variable ( a variable that represents a real number ), with two examples of domains that could be used for that predicate, such that the corresponding truth sets are not the same.**Geist:**Let*x*be a variable with domain**R**, and let*P*(*x*) be the predicate*x*≤ 5 →*x*^{2}≤ 25. F∈*D*the truth set.**Hint:**Break the domain up into smaller sets. For instance, a smart way would to be consider the following five sets:

{*x*:*x*< -5},{-5},{*x*: -5 <*x*< 5},{5},{*x*: 5 <*x*}

Determine whether*P*(*x*) is true on each of those five sets. Then use that information to determine the truth set of*P*.**Grueser:**Again let*x*be a variable with domain**R**, and this time let*Q*(*x*) be the predicate*x*^{2}≤ 25 →*x*≤ 5. F∈*D*the truth set.**Hint:**Same hint that I gave Geist.**Hisey:**Let*m*,*n*be variables with domain the set**Z**and let predicate*P*(*m*,*n*) be the sentence "*m*/*n*is an integer."- Give an example of a pair of integers
*a*,*b*such that*P*(*a*,*b*) is true. - Give an example of a pair of integers
*c*,*d*such that*P*(*c*,*d*) is true. - Give an example of a pair of integers
*e*,*f*such that*P*(*e*,*f*) is true and*P*(*f*,*e*) is false. - Give an example of a pair of integers
*g*,*h*such that*P*(*g*,*h*) is true and*P*(*h*,*g*) is also true.

- Give an example of a pair of integers
**Klions:**Again let*m*,*n*be variables with domain the set**Z**, but this time and let predicate*Q*(*m*,*n*) be the sentence "If*m*/*n*is an integer, then*n*/*m*is an integer. Which of the following pairs are in the truth set of*Q*(*m*,*n*) ? Explain.- (
*m*,*n*) = ( 3,6 ) - (
*m*,*n*) = ( 1,-1 ) - (
*m*,*n*) = ( 2,7 ) - (
*m*,*n*) = ( 0,5 ) - (
*m*,*n*) = ( 5,0 )

- (
**Korzan:**What does the symbol ∀ mean ?**Silveira:**Let*x*be a variable with domain the set*D*= { -2, -1, 0, 1, 2 }. Consider the universal statement ∀*x*∈*D*,*x*^{2}>*x*. Is the universal statement true ? Explain.**Strobel:**Now let*x*be a variable with domain the set**R**, and consider the universal statement ∀*x*∈**R**,*x*^{2}>*x*. Is this universal statement true ? Explain.**Weldon:**What does the symbol ∃ mean ?**Wohl:**Give an example of a true existentially quantified statement using a predicate*P*(*n*) with domain**Z**the set of integers. If possible, f∈*D*an example where if you use the same predicate and domain with a universal quantifier instead, the universally quantified statement is false.**Worthington:**Return to the predicate*P*(*x*) that Geist discussed. Su*p*ose we park a universal quantifier in front of the predicate to get the following statement*A*:

∀*x*∈**R**,*x*≤ 5 →*x*^{2}≤ 25.

Is statement*A*true ? Explain.**Bowsher:**Return to the predicate*Q*(*x*) that Grueser discussed. Su*p*ose we park a universal quantifier in front of the predicate to get the following statement*B*:

∀*x*∈**R**,*x*^{2}≤ 25 →*x*≤ 5.

Is statement*B*true ? Explain.**Carpenter:**What does the symbol ⇒ mean ?

Section 3.2 Predicates and Quantified Statements II

**Dewitt:**Let*A*be the universal statement, ∀*x*in*D*,*Q*(*x*). What is ~*A*?**Geist:**Let*A*be the statement "Every car in the Morton Hall Parking lot is silver". What is ~*A*?**Grueser:**Let*B*be the e*x*istential statement e*x*ists*x*in*D*such that*Q*(*x*). What is ~*B*?**Hisey:**Let*B*be the statement “There e*x*ists a car in the Morton Hall parking lot that is neon green”. What is ~*B*?**Klions:**Let*C*be the universal conditional statement ∀*x*in*D*, if*P*(*x*) then*Q*(*x*). What is ~*C*?**Korzan:**Consider the universal conditional statement*C*: ∀*x*∈,**R***x*≤ 5 →*x*^{2}≤ 25. Find the negation ~*C*. Which is true,*C*or ~*C*? Explain.**Silveira:**Consider the universal statement*S*. "Every prime number is odd." Find the negation, ~*S*. Which is true,*S*or ~*S*? Explain.**Strobel:**Rewrite the original*S*of Silveira as a universal conditional statement. Negate this new universal conditional statement version of*S*.**Weldon:**Let predicate*P*(*x*) be the sentence "*x*^{2}≥*x*." Find the truth value of*P*(*x*) for the following*x*-values:*x*= -2,-1,0,1,2. Are there any values of*x*for which*P*(*x*) is not true ? Explain.**Wohl:**Let*A*be the universal statement ∀*x*in*D*,*x*^{2}≥*x*. Is*A*true when domain*D*= {-2,-1,0,1,2} ? How about when*D*=? When**Z***D*=?**R****Worthington:**Let*C*be the statement ∀*x*∈,**R***x*≤ 5 →*x*^{2}≤ 25. Find ~*C*.**Bowsher:**Write the*contrapositive*,*converse*, and*inverse*of Worthington's statement*C*.**Carpenter:**Consider these five statements:- Worthington's original statement
*C* - The
*Negation*, ~*C* *Contrapositive*(*C*)*Converse*(*C*)*Inverse*(*C*)

- Worthington's original statement
**Cicigoi:**Let*S*be the statement ∀ integers*n*if 6/*n*is an integer, then*n*= 2 or*n*= 3. Write the negation ~*S*. Which is true,*S*or ~*S*? Explain.

Section 3.3 Statements with Multiple Quantifiers

**Changing the order of multiple quantifiers. **In each pair of statements, Statement *B* is obtained by switching the order the quantifiers of Statement *A*.

- Which of the statements are true ? ( might be one or both ) Explain.
- One of the statments is famous property of real numbers. Which statement, and what is the name of the property ? Explain.
- F∈
*D*the negation of any of any of the statements that are false.

**Geist:**- Statement A: ∀
*x*∈( ∃**R***y*∈(**Z***x*<*y*) ) - Statement B: ∃
*y*∈( ∀**Z***x*∈(**R***x*<*y*) )

- Statement A: ∀
**Grueser:**- Statement A: ∀
*x*∈( ∃**R***y*∈(**R***x*+*y*=*x*) ) - Statement B: ∃
*y*∈( ∀**R***x*∈(**R***x*+*y*=*x*) )

- Statement A: ∀
**Hisey:**- Statement A: ∀
*x*∈( ∃**R***y*∈(**R***x*+*y*= 0 ) ) - Statement B: ∃
*y*∈( ∀**R***x*∈(**R***x*+*y*= 0 ) )

- Statement A: ∀
**Klions:**- Statement A: ∀
*x*∈( ∃**R***y*∈(**R***x**y*=*x*) ) - Statement B: ∃
*y*∈( ∀**R***x*∈(**R***x**y*=*x*) )

- Statement A: ∀
**Korzan:**- Statement A: ∀
*x*∈* ( ∃**R***y*∈* (**R***x**y*= 1 ) ) - Statement B: ∃
*y*∈* ( ∀**R***x*∈* (**R***x**y*= 1 ) )

- Statement A: ∀

**Changing the domain in quantifiers. **Consider statement *S*: ∀ *x* ∈ *D* ( ∃ *y* ∈ *D* ( *x**y* > *y* ) ).

**Silveira:**Write the negation for*S*.**Strobel:**Is Statement*S*true when*D*=? Explain.**Z****Weldon:**Is Statement*S*true when*D*=? Explain.**R****Wohl:**Is Statement*S*true when*D*=**R**^{+}? Explain.

**Interchanging ∀ and ∃ in multiple quantifiers.**

**Worthington:**Consider Statement*A*and Statement*B*, which is obtained by interchanging ∀ and ∃ in the quantifiers of Statement*A*.- Statement
*A*: ∀*x*∈**R**^{+}( ∃*y*∈**R**^{+}(*y*<*x*) - Statement
*B*: ∃*x*∈**R**^{+}( ∀*y*∈**R**^{+}(*y*<*x*)

- Statement

**Interchanging x and y in the quantifiers. **Consider Statement

- Statement
*A*: ∀*x*∈( ∃**D***y*∈(**D***y*= 2*x*+ 1 ) - Statement
*B*: ∀*y*∈( ∃**D***x*∈(**D***y*= 2*x*+ 1 )

**Bowsher:**Let the domain*D*be the set. Is either of the statements**R***A*,*B*true ? Explain.**Carpenter:**Let the domain*D*be the set. Is either of the statements**Z***A*,*B*true ? Explain.

**Choosing correct order for symbols to create a true statement.**

**Cicigoi:**Consider the sentence ∀*x*∈ ____ ( ∃*y*∈ _____ (*y*= 1/*x*) ). Choose domains that work from this list of possible domains:**R**^{+},,**Q***. ( You can only use a domain once. )**Z****Dewitt:**You find thirteen cards scattered on the ground. They have the following symbols on them:

Assemble the cards in an order that makes a correct statement.**R****Z***x**y**y*= √*x*∀ ∃ ∈ ∈ ( ( ) )

3.4 Arguments with Quantified Statements

**Re-casting a rule in symbolic form.**

**Strobel:**In Section 3.4 of the book, the author presents a bunch of argument forms—some valid and some invalid—in symbolic form. These include Universal Modus Ponens, Universal Modus Tollens, Converse Error, Inverse Error, Universal Transitivity. But the very first “rule” presented at the very start of the section, called the rule of universal instantiation (p. 132), is not presented in symbolic form. Can you re-cast that rule in symbolic form ?

**Next Eight Students: Are these arguments valid or invalid ? Justify by citing a valid argument form or common error form. If possible, draw a Ve n diagram to illustrate.**

**Geist:**

All freshmen must take writing.

Caroline is a freshman.

∴ Caroline must take writing.**Grueser:**Same argument as Geist, but first rewrite it so that the first line is a universal conditional statement.**Hisey:**

If a product of two numbers is 0, then at least one of the numbers is 0.

For a particular number*x*, neither (2*x*+ 1) nor (*x*- 7) equals 0.

∴ The product (2*x*+ 1)(*x*- 7) is not 0.**Klions:**

All people are mice.

Mice are Mortal.

∴ All people are mortal.**Korzan:**

All healthy people eat an a*p*le a day.

Keisha eats an a*p*le a day.

∴ Keisha is a healthy person.**Silveira:**

All healthy people eat an a*p*le a day.

Herbert is not a healthy person.

∴ Keisha is a healthy person.**Weldon:**

All polynomial functions are differentiable.

All differentiable functions are continuous.

∴ All polynomial functions are continuous.**Wohl:**

Any sum of two rational numbers is rational.

The sum*r*+*s*is rational.

∴ The numbers*r*and*s*are both rational.

**Next Three Students: **Rewrite the argument so that its first statement is in the form of a universal conditional statement. (Notice that this can be done in two ways, because once you find a universal conditional statement that works, you can also use the contrapositive of that statement.) Are these arguments valid or invalid ? Justify by citing a valid argument form or common error form. If possible, draw a Ve*n* diagram to illustrate.

**Worthington:**

No good car is cheap.

A Yaris is cheap.

Therefore a Yaris is not good.**Bowsher:**

No good car is cheap.

A Jeep Patriot is not a good car.

Therefore a Jeep Patriot is cheap.**Carpenter:**

No perfect squares are prime.*n*is not a perfect square.

Therefore,*n*is prime.

**Last one!**

**Dewitt:**

(A) Use a diagram to show that the following argument form can have true premises and a false conclusion:

All dogs are carnivorous.

Aaron is not a dog.

Therefore, Aaron is not carnivorous.

(B) Is the following argument form valid ? Explain.

∀*x*, If*P*(*x*) then*Q*(*x*)

~*P*(*a*) for a particular*a*

∴ ~*Q*(*a*).

4.1 Direct Proof and Counterexample I: Introduction

**Hisey:**Is 0 even ? Explain**Klions:**Assume that*r*and*s*are particular integers (a) is 4*rs*even ? (b) is 6*r*+ 4*s*^{2}+ 3 odd ? Explain.**Korzan:**Can negative numbers be even ? Can they be odd ? Explain.**Silveira:**Are*even*&*odd*opposites ? That is, is every integer either even or odd ? Explain.**Strobel:**Is 1 prime ? Explain.**Weldon:**Can negative numbers be prime ? Can they be composite ? Explain.**Wohl:**Are*prime*&*composite*opposites ? That is, is every integer either prime or composite ? Explain.**Worthington:**If*r*and*s*are positive integers, is*r*^{2}+ 2*rs*+*s*^{2}is prime or composite or neither ? Explain.**Bowsher:**Is 2*n*^{2}- 5*n*+ 2 prime, composite, or neither ? Hint: try some values of*n*. Explain.**Carpenter:**Prove or disprove: For all integers*n*, if*n*is odd then (*n*- 1)/2 is odd. Hint: try some values of*n*.**Dewitt:**Let*D*= {1,2,3,4,5,6,7,8,9,10}. Prove by*exhaustion*: For each*n*in*D*, the number*n*^{2}-*n*+ 11 is prime.**Geist:**Releated problem: Now change the domain in Dewitt's statement to all positive integers, so that it reads as follows:

For every positive integer*n*, the number*n*^{2}-*n*+ 11 is prime.

Is the new statement true ? Explain.**Grueser:**Present the book's definition of*perfect square*. Then consider the book's definition of*consecutive integers*. Notice that the book's definition of consecutive integers does not name the numbers involved, and does not use any equations. Compare to the book's definition of perfect square. That definition names the number involved, and uses an equation to define the number's properties. Can you present the book's definition of consecutive integers, and also present your own version that names the numbers involved and uses an equation to define the property that the numbers have ?

4.2 Direct Proof and Counterexample II: Rational Numbers

**Klions:**Consider the list of numbers below. Which are rational, which are real ? Explain.- 0
- 0/5
- 5/0
- 753.86234
- 753.86234234234… (repeating decimal)

**Korzan:**Prove this statement: Every integer is a rational number.**Silveira:**Consider numbers whose decimal expansions terminate. Are they rational numbers or irrational irrational ? Explain with example.**Strobel:**Have some*pi*- Is 3.1416 (terminating decimal) a rational number ? Explain.
- Is π rational ? Explain.
- Does the decimal expansion for π ever repeat ? How do you know ? Explain.

**Weldon:**What is the*Zero Product Property*? Write it as a universal conditional statement. Then write the contrapositive of that universal conditional statement.**Wohl:**Why is the*Zero Product Property*introduced in this section on Rational Numbers ? Give an example of how the Zero Product Property is used in Section 4.2.**Worthington:**Prove: If*n*is odd then*n*^{2}+*n*is even.**Bowsher:**Prove: if*p*is any even integer and*q*is any odd integer, then (*p*+ 1)^{2}– (*q*- 1)^{2}is odd.**Carpenter:**Prove that the square of any rational number is rational.**Dewitt:**Prove that the product of any two rational numbers is rational.**Geist:**Consider this statement*S*: The quotient of any two rational numbers is rational.

Is Statement*S*true ? If it is true, then prove it. If*S*is not true, then modify it to make it a true statement, and prove the modified statement.**Grueser:**Consider this statement*S*: If*p*and*q*are rational, then (*p*+*q*)/2 is rational.

Is Statement*S*true ? If it is true, then prove it. If*S*is not true, then modify it to make it a true statement, and prove the modified statement.**Hisey:**Prove: ∀*a*,*b*∈**R**, if*a*<*b*, then*a*< (*a*+*b*)/2 <*b*.

4.3 Direct Proof and Counterexample III: Divisibility

**Korzan:**Do the symbols 5 / 7 and 5 | 7 mean the same thing ? If not, what do they mean ?**Silveira:**Do the symbols 2 / 6 and 2 | 6 mean the same thing ? If not, what do they mean ?**Strobel:**Frick and Frack are arguing. Frick says that 3 / 0 is undefined. Frack says that 3 | 0 is true. Who is right ? Explain.**Weldon:**Donkey and Elephant are arguing. Donkey says that 0 / 5 is zero. Elephant says that 0 | 5 is false. Who is right ? Explain.**Wohl:**Is 24 divisible by 3 ? Explain. Does 8 divide 40 ? Explain.**Worthington:**Does 6 | 42 ? Does 42 | 6 ? Does 6 | 6 ? Does 6 | (-24) ? Explain.**Bowsher:**For what*n*does 0 |*n*? For what*n*does*n*| 0 ? Explain.**Carpenter:**For what*n*does 1 |*n*? For what*n*does*n*| 1 ? Explain.**Dewitt:**The book defines the symbol*d*|*n*using an existentially quantified statement. Find the negation of that existentially-quantified statement. Compare your result to the book’s definition of symbol d ∤*n*. Why the difference ? Explain.**Geist:**- Prove that the sum of any three consecutive integers is divisible by 3.
**Hint:**Remember that in class on Monday, Jan 30, I remarked that a helpful way to think of a pair of consecutive integers is to say that they are a pair of numbers of the form*m*, (*m*+ 1). Do an analogous thing for three consecutive integers. - Does the result generalize ? That is, do you think the sum of any four consecutive integers is divisible by 4 ? Is the sum of any five consecutive integers divisible by 5 ? Explain

- Prove that the sum of any three consecutive integers is divisible by 3.
**Grueser:**Prove or disprove: If*ab*|*c*then*a*|*c*and*b*| c**Hisey:**Prove or disprove: If*a*|*bc*then*a*|*b*or*a*| c**Klions:**Prove or disprove: If*a*|*b*then*a*^{2}|*b*^{2}

4.4 Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem

**Silveira:**Find 50*div*7 and 50*mod*7 Show the corresponding*n*=*dq*+*r*equation**Strobel:**Find -50*div*7 and -50*mod*7 Show the corresponding*n*=*dq*+*r*equation**Weldon:**Find 56*div*7 and 56*mod*7 Show the corresponding*n*=*dq*+*r*equation**Wohl:**If*c*is an integer such that*c**mod*13 = 5, then what is 6*c**mod*13? In other words if division of*c*by 13 gives a remainder of 5, what is the remainder when 6*c*is divided by 13? Explain using*n*=*dq*+*r*equations.**Worthington:**What is the*parity property*, and what is the*parity property theorem?*Explain**Bowsher:**Prove that the square of any integer has the form 4*k*or 4*k*+ 1 for some integer*k*- The goal is to prove that the product of any three consectutive integers is disible by 3 using two different methods.
**Carpenter:**Use brute force: let the three consecutive integers be*m*, (*m*+ 1), (*m*+ 2). Find their product, and show that it is divisible by 3.**Dewitt:**Instead of using brute force, use the quotient-remainder theorem with*d*= 3.

**Geist:**Use quotient remainder theorem with*d*= 3 to prove that the product of any two consecutive integers has form 3*k*or 3*k*+ 2**Grueser:**Use quotient remainder theorem with*d*= 3 to prove that the square of integer has form 3*k*or 3*k*+ 1**Hisey:**Rewrite the expression |*x*- 1|/(*x*- 1) in three cases:- case 1:
*x*- 1 > 0 - case 2:
*x*- 1 = 0 - case 3:
*x*- 1 > 0

*f*(*x*) = |*x*- 1|/(*x*- 1) as a piecewise-defined function- case 1:
**Klions:**Prove ∀*r*,*c*∈with**R***c*≥ 0, if -*c*≤*r*≤*c*, then |*r*| ≤*c***Korzan:**Verify the triangle inequality in these cases: (*x*,*y*) = (5,-3),(5,3),(-5,3),(-5,-3),(-1,0),(0,0)

Sections 4.4 and 4.6

**Dewitt:**Use the quotient-remainder theorem with*d*= 3 to prove that product of any three consectutive integers is disible by 3 .**Geist:**Use quotient remainder theorem with*d*= 3 to prove that the product of any two consecutive integers has form 3*k*or 3*k*+ 2**Grueser:**Use quotient remainder theorem with*d*= 3 to prove that the square of integer has form 3*k*or 3*k*+ 1**Hisey:**Rewrite the expression |*x*- 1|/(*x*- 1) in three cases:- case 1:
*x*- 1 > 0 - case 2:
*x*- 1 = 0 - case 3:
*x*- 1 > 0

*f*(*x*) = |*x*- 1|/(*x*- 1) as a piecewise-defined function- case 1:
**Klions:**Prove ∀*r*,*c*∈with**R***c*≥ 0, if -*c*≤*r*≤*c*, then |*r*| ≤*c***Korzan:**Verify the triangle inequality in these cases: (*x*,*y*) = (5,-3),(5,3),(-5,3),(-5,-3),(-1,0),(0,0)

**Carpenter:**The goal is to prove statement*S*: ∀*n*∈( If**Z***n*^{2}is odd then*n*is odd.)- Write the contrapositive of
*S*. - Prove the contrapositive.

- Write the contrapositive of
**Silveira**Consider the following statement:

"If a product of two positive real numbers is greater than 100 then at least one of the numbers is greater than 10."

In symbols, this would be written:

Statement*S*: ∀*x*,*y*∈**R**^{+}( If*xy*> 100 then (*x*> 10 or*y*> 10).)- Write the contrapositive of
*S*. - Prove the contrapositive.

- Write the contrapositive of
**Strobel:**Consider the following statement: "There is no least positive real number"- It is probably simplest to first fine the negation of statement
*S*in prose. Do that. - Then write the negation ofstatement
*S*as using variables and quantifiers. - Prove
*S*using the method of contradiction. That is, assume that ~*S*is true, and then reach a contradiction.

- It is probably simplest to first fine the negation of statement
**Weldon:**The goal is to prove the following Statement*S*: For any integer*n*, the number*n*^{2}- 2 is not divisible by 4.- Write the negation of
*S*. - Prove that
*S*is true using a proof by contradiction. (That is, assume that ~*S*is true, and then reach a contradiction.) (Hint: There are two cases:*n*even and*n*odd)

- Write the negation of

Section 5.1 Sequences

**Bowsher:**5.1 # 21**Carpenter:**5.1 # 22**Dewitt:**5.1 # 24**Geist:**5.1 # 25**Grueser:**5.1 # 26**Hisey:**5.1 # 44**Klions:**5.1 # 45**Korzan:**5.1 # 50**Silveira**Compute 6!/4! Show the details clearly.**Strobel:**Compute 6!/5! Show the details clearly.**Weldon:**Compute 6!/1! Show the details clearly.**Wohl:**Compute 6!/1! Show the details clearly.**Worthington:**Compute*n*!/(*n*- 2)! Show the details clearly.

**Useful Symbols: **~ ∧ ∨ → ⇒ ∴ ∀ ∃ ∈ ≤ ≥ ε *x*^{2}

( page maintained by Mark Barsamian, last updated February 14, 2017 )