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.

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: Not all triangles 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 negation 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

Write the negation of p

(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= {studentswho study math }∧¿

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.

(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∈ {eas t 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 FF T FF F F

p q p∨qT T TT F TF T TF F F

Practice

Determine if the compound propositions p∧q∧p∨q are 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⋀ q , p∨q ,∧¬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------------------------------------------------------->

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.) p: x is a factor of 20. q: x is a factor of 50.

Determine 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

p q p ∨ qT T FT F TF T TF F F

ANSWER for 11 Part b

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

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

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

(iv.) Every integer 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 not an odd number and is 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 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 compound 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

Practice:

Construct a truth table for each of the following compound propositions:

(1) p∧¬( p∨q) (2) ¬ ( p∧q )∧¬(p∨q) (3) (¬ p∨¬ q¿∧¬(r∨q)

Answers:

(1) p∧¬( p∨q)

p q p∨q ¬( p∨q) p∧¬( p∨q)

T TT FF TF F

(2) ¬ ( p∧q )∧¬(p∨ q)

p Q p∧q p∨q ¬ ( p∧q ) ¬( p∨q) ¬ ( p∧ q )∧¬(p∨q)

T TT FF TF F

(3) (¬ p∨¬ q¿∧¬(r∨q)

p q R ¬ p ¬ q (¬ p∨¬ q¿

r∨q ¬(r∨q) (¬ p∨¬ q¿∧¬(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 p ⇒ qT T TT F FF T TF F T

Practice:

Write out a complete sentence representing the implication ( p⇒q) for each pair of propositions.

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

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

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

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

Construct a truth table for the following compound propositions

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

(8.) ¬ ( p⇒q )⇒ (¬ q⇒ p)

ANSWERS

(5. ) ¬ p⇒q

p q T TT FF TF F

(6) (p∧q¿⇒ q

p q T TT FF TF F

(7.) ¬ q∧(¬ p⇒ q)

p q T TT FF TF F

(8.) ¬ ( p⇒q )⇒ (¬ q⇒ p)

p q T TT FF TF F

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 denoted p⇔q.

Note:

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

Truth Table for Equivalence

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 is not doing

math.

Construct truth tables to show the following compound propositions are equivalent.

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

(12.) (¬ p⇒q )=( p∧¬q) (13.) (¬ p ⇔ ¬q )=¿ ¬( p ⇔q)

p q p ⇔q

T T TT F FF T FF F T

Logic Unit Exam Review

Copy the problem and answer the questions that follow.

1. Three propositions are defined as follows:

p: The oven is working.

q: The food supply is adequate.

r: The visitors are hungry.

(a) Write one sentence, in words only, for each of the following logic statements.

(i) q r p(2)

(ii) r (p q)(2)

(b) Write the sentence below using only the symbols p, q and logic connectives.

"If the oven is working and the food supply is adequate then the oven is working or the food supply is adequate."

(2)

(c) A tautology is a compound statement which is always true. Use a truth table to determine whether or not your answer to part (b) is a tautology.

Hint: Begin by writing the first two columns of your truth table in the following format:

p q

T T

T F

F T

F F(3)

(Total 9 marks)

ANSWER1. (a) (i) “The food supply is adequate and the visitors are hungry but the

oven is not working,” (or equivalent statement).

(ii) “Either the oven is working and the food supply is adequateor the visitors are not hungry,” (or equivalent statement).

(b) (p q) (p q)

(c)p Q (p q) (p q) (p q) (p q)

T T T T T

T F F T T

F T F T T

F F F F T

Therefore, (p q) (p q) is a tautology

Practice

2. [(p q) p] q

(a) Complete the truth table below for the compound statement above.

p q p q (p q) p [(p q) p] q

T TT FF TF F

(b) Explain the significance of your result.

(Total 4 marks)

Answer

2. (a)p Q p q (p q) p [(p q) p] qT T T T TT F F F TF T F F TF F T F T

(b) It is a tautology (or equivalent). The statement is valid.

Practice

3. Consider the following logic statements:

p: the train arrives on timeq: I am late for school

(a) Write the expression p q as a logic statement.

(b) Write the following statement in logic symbols:

"The train does not arrive on time and I am not late for school."

(c) Complete the following truth table.

p q p q p q p q

T T F F F FT F F T T –F T T F – –F F T T T T

(d) Are the two compound propositions (p q) and ( p q) logically equivalent?

(Total 8 marks)

Answer

3. (a) If the train arrives on time then I am not late for school.

(b) ¬p ¬q

(c) missing elements are FTF

(d) The implications are not logically equivalent.

Practice

4. (a) (i) Complete the truth table below.

p q p q (p q) p q p q

T T F F

T F F T

F T T F

F F T T

(ii) State whether the compound propositions (p q) and pq are equivalent.

(4)

Consider the following propositions.

p: Amy eats sweetsq: Amy goes swimming.

(b) Write, in symbolic form, the following proposition.

Amy either eats sweets or goes swimming, but not both.(2)

(Total 6 marks)

Answer4. (a) (i)

p q p q (p q) p q p q

T T T F F F F

T F F T F T T

F T F T T F T

F F F T T T T

(ii) Yes.

(b) p q.

Practice

5. Consider the following statements.

p: students work hardq: students will succeed

(a) Write the following proposition in symbols using p, q and logical connectives only.

If students do not work hard, then they will not succeed.

(b) Complete the following truth table, relating to the statement made in part (a), and decide whether the statement is logically valid.

p q

T TT FF TF F

(Total 8 marks)

Answer5. (a) ¬p ¬q

(b)p q ¬p ¬q ¬p ¬qT T F F TT F F T TF T T F FF F T T T

Practice

6. Consider each of the following statements:

p: Alex is from Uruguayq: Alex is a scientistr: Alex plays the flute

(a) Write each of the following arguments in symbols:

(i) If Alex is not a scientist then he is not from Uruguay.

(ii) If Alex is a scientist, then he is either from Uruguay or plays the flute.

(3)

(b) Write the following argument in words:

r (q p)(3)

(c) Construct a truth table for the argument in part (b) using the values below for p, q, r and r. Test whether or not the argument is logically valid.

p q r r

T T T FT T F TT F T FT F F TF T T FF T F TF F T FF F F T

(4)(Total 10 marks)

Answer6. (a) (i) ¬q ¬p

(ii) q (p r)

(b) If Alex does not play the flute then it is not true that he is a scientistor from Uruguay.ORIf Alex does not play the flute then he is neither a scientist norfrom Uruguay.

(c)q p ¬(q p) ¬r ¬(q p)

T F TT F FT F TT F FT F TT F FF T TF T T

Not a logically valid argument

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⇒¬ q

T T F F TT F F T TF T T F FF F T T T

p q ¬ p ¬ q ¬ q⇒¬ p

T 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.

(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 value 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.Scenario: You are working on number 14.

Practice

For each of the following problems you are given a statement that you are to then rewrite as an implicative statement. Write the implicative statement and then also write the converse, inverse, and contrapositive of each statement.

Ex.) People who go to White SUCKS games have a horrible time.

ANSWERImplication: If you go to a White Sox game, then you will have a horrible time.

Converse: If you have a horrible time, then you are at a White Sox game.

Inverse: If you do not go to a White Sox game, then you will not have a horrible time.

Contrapositive: If you do not have a horrible time, then you are not at a White Sox game.

Practice Same directions as above

(1) All IB students love to take logic quizzes.

(2) High school students who play baseball are naturally smarter than all other forms of human life.

(3) Female IB students at Washington high school always wish for food as all rewards.

Answers

(1)Implicative Statement: If you are an IB student, then you love to take logic quizzes.

Converse: If you love to take logic quizzes, then you are an IB student.

Inverse: If you are not an IB student, then you do not love to take logic quizzes.

Contrapositive: If you do not love to take logic quizzes, then you are not an IB student.

(2)Implicative Statement: If you are a high school student who plays baseball, then you are naturally smarter than all other forms of human life. Except if you live in Slag Valley.

Converse: If you are naturally smarter than all other forms of human life, then you are a high school student who plays baseball.

Inverse: If you are not a high school student who plays baseball, then you are not naturally smarter than all other forms of human life.

Contrapositive: If you are not naturally smarter than all other forms of human life, then you are not a high school student who plays baseball.

(3)Implicative Statement: If you are a female IB student at Washington high school, then you wish for food as all your rewards.

Converse: If you wish for food as all your rewards, then you are a female IB student at Washington high school.

Inverse: If you are not a female IB student at Washington high school, then you do not wish for food as all your rewards.

Contrapositive: If you do not wish for food as all your rewards, then you are not a female IB student at Washington high school.