Unit 2: Logic

Bell-Ringer: Day 1

Which of the following statements is true or false, write each statement out.

1.) Washington high school is mostly made of brick.

2.) The MATRIX is the best movie of all time.

3.) Earth has a Moon.

4.) America is the best country to live in on planet earth.

5.) World war 2 is over.

Color Key:

Green = Definitions Yellow = Regular highlighting

Mathematical Logic: deals with the conversion of worded statements into symbols, and how we apply rules of deduction to them.

Proposition: is a statement which may be true or false.

Indeterminate Proposition: a proposition whose answer would not be the same for all people.

Example.) The MATRIX is the best movie of all time.

Truth Value: is used to refer to whether a proposition is true or false.

Practice:

Which of the following are propositions? If they are, state if they are true, false, or indeterminate? Yes, write them all out.

6.) 20÷4=80 7.) 25×8=200 8.) Where is my pen? 9.) Your eyes are blue.

How do we represent propositions? Do we write out the whole statement every time? Hell no!

This is math we always try to simplify things. Take a guess.

Yeah that's right, with variables. We mostly use the letters p, q, and r. However, you can use any letter.

Logic Notation for Propositions ↓

examples.) p: It always rains on Tuesdays Not sure why, but a

q: 37 + 9 = 46 colon is used not an r: x is an even number equal sign.

So another question. How do you think, in mathematical logic, we refer to something like A' ? You know, the complement of set A from the Venn diagrams we spent a lot of time working on.

Negation: The negation of a proposition p is "not p" and is denoted ¬p . The truth value of ¬p is the opposite of the truth value of p.

Practice:

Write the negation for each of the following and note if the proposition or the negation is true.

p: All triangles have congruent sides.

q: 90÷9=10

r: √10 is an irrational number.

a: 16 - 3 = 20

b: anything divided by zero is zero.

c: All right angles are equal.

d: If a triangle has 3 congruent sides then it has 3 congruent angles.

e: x≥12

f: 4<x≤15

g: All Z+¿ ¿are also Q'

Answers:

¬p: All triangles do not have congruent sides. The negation is true.

¬q: 90÷9≠10 The original proposition is true.

¬r: √10 is not an irrational number. The original proposition is true.

¬a: 16−3≠20 The original proposition is true.

¬b: Anything divided by zero is not zero. The negation is true.

¬c: All right angles are not equal. The original proposition is true.

¬d: IF a triangle has 3 congruent sides then it does not have 3 congruent angles.

The original proposition is true.

¬e: x<12 Indeterminate

¬ f : x≤4∧x>15 Indeterminate

¬g : All Z+¿ are not alsoQ' ¿ The negation is true.

Day2

Practice more negation problems

(1.) p: x≥15 for x∈Z+¿¿ (2.) p: x is a dog for x∈ {cats , rats ,dogs , deer }

(3.) p: x≥10 for x∈Z (4.) p: x is a male athlete for x∈ {athletes }

(5.) p: x is a female athlete for x∈{females }

Answers:

(1.) ¬ p : 1≤x ≤14 for x∈Z+¿ ¿

(2.) ¬ p : x∈ {cats , rats ,deer }

(3.) ¬ p : x<9 for x∈Z

(4.) ¬ p : x is a female athlete

(5.) ¬ p : x is a female non-athlete

What are we doing Next?

MORE VENN DIAGRAMS!!!!!Notes:

We can use Venn diagrams for propositions that have variables where the truth value can change depending on the value of the variable

U is the universal set of all the values that the variable x may take.

P is the truth set of the proposition p, or the set of values of x∈U for which p is true.

P' is the truth set of ¬ p

See Drawing on the Board

Ex.) For U={x|0< x<10 , x∈N } and proposition p: x is a prime number, find the truth sets of p and ¬ p . Draw a Venn diagram with this information.

SOLUTION WITH VENN DIAGRAM------------------>

Practice:

(6.) Suppose U={students∈ year12 }M= {studetnswho studymath }∧¿

G={Students who play guitar }

Draw a Venn Diagram to represent each of these statements

a.) All students that study Math play guitar.

b.) None of the students who play guitar study math.

c.) No one that plays guitar does not study math.

VENN DIAGRAMS ------------------------------------>

(7.) Represent U={x|7≤ x<14 , x∈N }and p: x < 11 on a Venn diagram. List the truth set of ¬ p.

Day 3

Bell Ringer:

Write out each of the following propositions and write the negation for each one of them. You have 8 min. Then I will collect it.

(1.) p: x≥9 for x∈Z+¿¿

(2.) p: x<−5 for x∈Z−¿¿

(3.) p: If a triangle is isosceles then it has 3 congruent sides.

(4.) p: x is a dog for x∈ {animals }

(5.) p: x is a male student for x∈ {east side people }

Compound Propositions

Conjunction: is 2 propositions that are joined with the word "and".

denoted: p∧q

Disjunction: is 2 propositions that are joined with the word "or"

denoted: p∨q

Truth values for each compound proposition is listed below. We call these Truth Tables

Conjunction: Disjunction:

For a Venn Diagrams :

Conjunction Disjunction

The truth set of p∧q is P⋂Q The truth set of p∨q is P⋃Qthe region where both p and q are true. the region where p or q or both are true.

Venn Diagrams on Board--------------------------------------------------------------------------------------->

p q p∨qT T TT F TF T TF F F

p q p∧qT T TT F FF T FF F F

Practice

Determine if the compound propositions p∧q∧p∨q is true or false. Yes, write everything out.

(6.) p: 30 is a multiple of 5 q: 30 is a multiple of 4

(7.) p :−10≥−9q :10≥9

(8.) p :20÷5−9 ∙2=14 q :32 (8 ∙2−10÷2 )÷3=9

(9.) For U={x∨8≤ x≤15 , x∈Z } consider the propositions

p: x is a multiple of 3 q: x is an odd number

a.) Illustrate the truth sets for p and q and a Venn diagram.

b.) Use your Venn diagram to find the truth set for

i.) ¬q ii.) p∧q iii) p∨q iv.) ¬ p

Day 4

Bell-Ringer:(1.) p: x≥16 for x∈Z+¿¿

(2.) p: x←9 for x∈Z−¿¿

(3.) p: x≥−6 for x∈Z

(4.) p: x is all staplers that do not belong to Mr. Alvarez. for x∈ {office supplies}

(5.) p: x is all people that do not get distracted by small animals. for x∈ {Schreibers}

Venn Diagram Practice:(6.) Suppose U={x∨8<x ≤14 , x∈N }

p: x < 11q: multiples of 2(a.) draw a Venn diagram with this information.(b.) List the truth set of ¬ p and ¬q

New Compound Proposition

Exclusive Disjunction: The exclusive disjunction is true when ONLY one of the propositions is true.

Denoted: p ∨ q

Example: p: Sally ate cereal for breakfast.

q: Sally ate toast for breakfast.

p ∨ q: Sally ate cereal or toast for breakfast, but did not eat both.

Exclusive Disjunction Truth Table

Venn Diagram------------------------------------------------------->

p q p ∨ qT T FT F TF T TF F F

Practice:

Write the exclusive disjunction for the following pairs of propositions.

(7.) p: I will go to Petes today. q: I will go to the ATM today.

(8.) : x is a factor of 20. q: x is a factor of 50.

Write the exclusive disjunction for the following propositions and find the truth value of p ∨ q.

(9.) p: 20 is odd q: 25 is a multiple of 5.

(10.) p: 5.7 ∈Z q: 9∈N

(11.) For U = {x|2≤ x<13 , x∈Z }, consider the propositions

p: x is an even number

q: x is a number divisible by 3 that produces no remainder.

a. ) Illustrate the truth sets for p and q on a Venn diagram

b.) Write down the meaning of these propositions in complete sentences

i.) p∧q ii.) p∨q iii) p ∨ q iv) ¬( p∧q)

c.) Use your Venn diagram to find the truth sets for

i.) p∧q ii.) p∨q iii) p ∨ q

ANSWER for 11 Part b

(i) Every number between 1 and 13 that is both even and that when divided by 3 produces no remainder.

(ii) Every number between 1 and 13 that is even or that when divided by 3 produces no remainder.

(iii) Every number between 1 and 13 that is either even or that when divided by 3 produces no remainder, but not both.

(iv.) Every number between 1 and 13 that is not both even and that when divided by 3 produces no remainder.

Answers for 11 Part C

i.) p∧q: {6 ,12 }

ii.) p ∨ q: {2, 3, 4, 6, 8, 9, 10, 12}

iii.) p ∨ q : {2, 10, 4, 8, 3, 9}

iv.) ¬( p∧q): {2, 10, 4, 8, 3, 9, 5, 7, 11, }

12.) For U = {x|0≤x<12 , x∈Z }, consider the propositions

p: x is a multiple of 4

q: x is an odd number

a. ) Illustrate the truth sets for p and q on a Venn diagram

b.) Write down the meaning of these propositions in complete sentences

i.) ¬ p ii.) ¬q iii) p ∨ q iv) ¬ p∧q

c.) Use your Venn diagram to find the truth sets for

i.) p∧q ii.) p ∨ q iii) ¬( p∨q) iv) ¬(¬ p∧¬q)

Answers for 12 Part B

i.) Every integer between -1 and 12 that is not a multiple of 4.

ii.) Every integer between -1 and 12 that is not odd.

iii.) Every integer between -1 and 12 that is either an odd number or a multiple of 4, but not both.

iv.) Every integer between -1 and 12 that is not a multiple of 4 and is an odd number.

Answers for 12 Part C

i.) p∧q: ϕ

ii.) p ∨ q: {4, 8, 1, 3, 7, 5, 9, 11}

iii.) ¬( p∨q): {0, 2, 6, 10}

iv.) ¬(¬ p∧¬q): {4, 8, 1, 3, 7, 5, 9, 11}

IN CLASS ASSESSMENT:

Let u={4 ≤ x ≤14 , x∈Z } p: x is a prime numberq: is an odd number

(1.) Illustrate the truth sets for p and q on a Venn diagram.

(2.) Write down the meaning of each proposition given.

a.) ¬ p b.) p ∨ q

(3.) Use your Venn diagram to find the truth set for the exclusive disjunction.

MORE PRACTICE

(13.) For U = {x|5≤ x≤17 , x∈Z }, consider the propositions

p: x is an odd number

q: x is a multiple of 3

a. ) Illustrate the truth sets for p and q on a Venn diagram

b.) Write down the meaning of these propositions in complete sentences

i.) ¬ p ii.) ¬q iii) p ∨ q iv) ¬ p∧q

c.) Use your Venn diagram to find the truth sets for

i.) p∧q ii.) p ∨ q iii) ¬( p∨q) iv) ¬(¬ p∧¬q)

ANSWERS for 13 Part B

b.) ¬ p: Every integer between 4 and 18 that is not an odd number.

¬q : Every integer between 4 and 18 that is not a multiple of 3.

p ∨ q: Every integer between 4 and 18 that is either an odd number or a multiple of 3, but not both.

¬ p∧q : Every integer between 4 and 18 that is both not an odd number and a multiple of 3.

(14.) For U = {x|0<x≤11 , x∈Z }, consider the propositions

p: x is numbers that have a t in their spelling.

q: x is a prime number.

a. ) Illustrate the truth sets for p and q on a Venn diagram

b.) Write down the meaning of these propositions in complete sentences

i.) p∧q ii.) p∨q iii) p ∨ q iv) ¬( p∧q)

c.) Use your Venn diagram to find the truth sets for

i.) p∧q ii.) ¬(p ∨ q) iii) ¬( p∨q) iv) ¬( p∧¬q )

Answers for 14 Part B

i.) p∧q: Every integer between 0 and 12 that is a number with a t in its spelling and is prime.

ii.) p∨q: Every integer between 0 and 12 that is a number with a t in its spelling or is prime.

iii.) p ∨ q: Every integer between 0 and 12 that is either a number with a t in its spelling or is prime, but not both.

iv.) ¬( p∧q): Every integer between 0 and 12 that is not both a number with a t in its spelling and is prime.

SO NOW WHAT? OH YOU WANT TO KNOW. OK.

TRUTH TABLES!Complete summary of the truth values of all our compound propositions

Negation Conjunction Disjunction Exclusive Disjunction

p q ¬ p p∧q p∨q p ∨ qT T F T T FT F F F T TF T T F T TF F T F F F

New Definitions

Tautology: is a compound proposition where all the values in its truth table column are true.

Logical Contradiction: a compound proposition where all the values in its truth table column are false.

Logically Equivalent: is 2 propositions that have the same truth table column.

Practice:

Construct a truth table for the following propositions.

(1.) ¬ p∧q (2.) ¬ (p ∨ q) (3.) ¬ p∧¬q (4.) ¬( p∨q)

Answers---------------------------------------------------------------------------------

Bell Ringer:

Construct a truth table for the following proposition.

(1.) p∧¬(r∨¬q)

p q r ¬q r∨¬q ¬(r∨¬q) p∧¬(r∨¬q)T T TT T FT F TT F FF T TF T FF F TF F F

New Compound Proposition

Implication

If two proposition can be linked with “if …., then…..” then we have an implication. The implicative statement “if p then q” is written p ⇒ q and reads “p implies q”. p is called the antecedent and q is called the consequent.

Example: Write the implication p ⇒ q for the following propositions.

p: It will rain on Sunday.

q: The Bears will win.

p ⇒ q: If it rains on Sunday, then the Bears will win.

Truth Table

p q Scenario p ⇒ qT T It rains on Sunday, and the Bears win. T

T F It rains on Sunday, and the Bears lose. FF T It does not rain on Sunday, and the Bears win. TF F It does not rain on Sunday, and the Bears lose. T

Practice:

Write out the implication for each pair of propositions.

(2.) p: The sun is shining q: I will play baseball

(3.) p: x is a multiple of 8 q: x is an even number

(4.) p: There is a rake in the garage q: I will clean up the leaves

(5.) p: Mr. Alvarez wears red and blue q: Mr. Alvarez is Superman

Construct a truth table for the following propositions

(6.) ¬ p⇒q (7.) (p∧q ¿⇒ q (8.) ¬q∧(¬ p⇒ q)

(9.) ¬ (p⇒q )⇒ (¬q⇒ p)

Equivalence: If 2 propositions are linked with “if and only if" then we have an equivalence. The equivalence "p if and only if q" is denotedp⇔q.

Note:

p⇔q Is logically equivalent to the compound proposition ( p⇒q )∧(q⇒ p)

Truth Table for Equivalence

p q p⇔q

T T TT F FF T FF F T

Practice

(9.) Write out the following compound propositions in terms of the 2 propositions given (yes, that means a sentence). Also, write a scenario that would make that proposition false.

p: Cars beep their hornsq: Mr. Alvarez does math

(a.) p⇒q (b.)q⇒ p (c.) p⇔q (d.) ¬ p⇔q

Answer Part (a.)

p⇒q : If cars beep their horns, then Mr. Alvarez does math.False case: Cars are beeping their horns, but Mr. Alvarez isn't doing math.

(10.) Construct a truth table that shows p⇔q=( p⇒q )∧(q⇒ p)

Construct truth tables for the following propositions.

(11.) p ∨ q = ¬( p⇔q) (12.) (¬ p⇒q )=( p∧¬q)

(13.) (¬ p⇔¬q )=¿ ¬( p⇔q) (14.)

CONVERSE, INVERSE and CONTRAPOSITIVE

Converse: the converse of the implication p⇒q is the statement q⇒ p

Truth Table for the converse

p q q⇒ p

T T TT F TF T FF F T

Inverse: the inverse of the implication p⇒q is the statement ¬ p⇒¬q

Truth Table for the inverse

Same truth table as the converse. Thus, they are logically equivalent.

Contrapositive: the contrapositive of the implication p⇒q is the statement ¬q⇒¬ p

Truth Table for the Contrapositive

Same truth table as the implication p⇒q. Thus, the implication and its contrapositive are logically equivalent.

p q ¬ p ¬q ¬ p⇒¬qT T F F TT F F T TF T T F FF F T T T

p q ¬ p ¬q ¬q⇒¬ pT T F F TT F F T FF T T F TF F T T T

Practice:

Write the converse and inverse for each implication given. Also write a scenario for each converse and inverse that makes it false.

(1.) If John owns a mustang, then he is a cool guy.

(2.) If Mr. Alvarez keeps playing the Illinois lottery, then Mr. Alvarez will continue to be sad.

(3.) If you’re obscenely confident, then your name is John Wayne.

(4.) If you get killed by John Wayne on XBOX Live, then Mr. Alvarez is tearing it up on Halo 4.

Answers:

(1.) Converse: If you're a cool guy, then you’re John and have a mustang.

False: John doesn't own a mustang and he is a cool guy.

Inverse: If john doesn't own a mustang, then he is not a cool guy.

False: John doesn't own a mustang and he is a cool guy.

(2.) Converse: If Mr. Alvarez continues to get sad, then he keeps playing the Illinois lottery.

False: Mr. Alvarez is not playing the Illinois lottery and he keeps getting sad.

Inverse: If Mr. Alvarez does not keep playing the Illinois lottery, then he will not keep getting sad.

False: Mr. Alvarez is not playing the Illinois lottery and he keeps getting sad.

(3.) Converse: If your name is John Wayne, then you're obscenely confident.

False: You're not obscenely confident and your name is John Wayne.

Inverse: If you're not obscenely confident, then you're not John Wayne.

False: You're not obscenely confident and you're name is John Wayne.

(4.) Converse: If Mr. Alvarez is tearing it up on Halo 4, then you got killed on XBOX live by John Wayne.

False: You didn't get killed on XBOX live by John Wayne and Mr. Alvarez is tearing it up on Halo 4.

Inverse: If you didn't get killed on XBOX Live by John Wayne then Mr. Alvarez is not tearing it up on Halo 4.

False: You didn't get killed on XBOX live by John Wayne and Mr. Alvarez is tearing it up on Halo 4.

Contrapositive Practice

Write down the contrapositives for each statement

(Example.) All teachers drive blue cars

1st write the implicative statementp⇒q: If you're a teacher, then you drive a blue car.orp⇒q: If a person is a teacher, then he or she drives a blue car.

2nd write Down p and q:p: You are a teacher q: You drive a blue car orp: A person is a teacher q: A person that drives a blue car

3rd write down the negation of p and q ¬ p: You are not a teacher or ¬ p: A person is not a teacher ¬q : You do not drive a blue car or ¬q : A person does not drive a blue car

Lastly, build the contrapositive from the components in the 3rd step.¬q⇒¬ p: If you do not drive a blue car, then you are not a teacher.or¬q⇒¬ p :If a person does not drive a blue car then the person is not a teacher

Practice

Write down the contrapositives for each statement

(5.) All car dealerships overcharge for repairs.(6.) All players named John Wayne are dirty screen watchers.(7.) All Peoria soccer players are lightning fast.(8.) John Wayne never backs down from a cowardly sniper.(9.) John Wayne always causes fear and paranoia in his less skilled opponents.

For each of the following implications given write down the converse, inverse and contrapositive implications. Determine the truth values for the converse inverse and contrapositive for each scenario described.

(10) If you are a teenage girl, then you love the band One Direction.Scenario: You’re a teenage girl and you don’t love the band One Direction.

(11) If you don’t like pizza, then you are not human.Scenario: You love pizza and you are human.(12) The MATRIX never releases a human without a fight.Scenario: You are not in the matrix and you were not released without a fight.

(13) Fabian will never take down John Wayne.Scenario: Fabian was playing halo and killed John Wayne(14) All problems in Math Studies 2 incorporate John Wayne somehow.