Inductive Reasoning and Conjectures

Multiple Choice

Which of the following is a counterexample to the following conjecture? If $$x^2\ =\ 4$$ , then x = 2

• x = 4

• x = -2

• x = 1

• x = -4

Multiple Choice

How many squares are in Step 5?

• 30

• 31

• 32

• 33

• 34

Multiple Choice

Where would the next red square be?

• bottom left

• top right

• top left

• bottom right

Multiple Choice

Complete each conjecture:

The product of two even numbers is ___?

• odd

• even

• could be either odd or even

• 0

​Inductive Reasoning, Conjectures, and Counterexamples

By Nicole O'Donnell

Multiple Choice

Inductive Reasoning means...

• Guessing
• Testing and observing patterns to make conjectures
• Explaining why
• Ura nok seblu!

Multiple Choice

Inductive Reasoning means...

• Guessing
• Testing and observing patterns to make conjectures
• Explaining why
• Ura nok seblu!

Multiple Choice

Inductive Reasoning means...

• Guessing
• Testing and observing patterns to make conjectures
• Explaining why
• Ura nok seblu!

Multiple Choice

How would you describe this pattern's rule?

16, 11, 6, 1

• Add 5

• Subtract 3

• Subtract 5

• Subtract 6

Multiple Choice

What number belongs in the blank?

18,24,___,36,42

• 28

• 31

• 30

• 35

Multiple Choice

Complete the conjecture.
The sum of two negative numbers is ___________.

• positive
• negative
• odd
• even

Multiple Choice

Find the next number in the sequence.
1, -1, 2, -2, 3, ___

• 4
• -4
• 3
• -3

Multiple Choice

Which is a counterexample to the following statement?

If an angle is obtuse, then it is $$125\degree$$ .

• $$90\degree$$  

• $$57\degree$$  

• $$160\degree$$  

• $$180\degree$$  

Multiple Choice

Which of the following provide a counterexample to the claim:

"If two angles are supplementary, then they are not congruent."

• $$40\degree\ and\ \ 140\degree$$

• $$80\degree\ and\ \ 80\degree$$

• $$30\degree\ and\ \ 150\degree$$

• $$90\degree\ and\ \ 90\degree$$

Multiple Choice

Which is a counterexample to the following statement?

If you add two numbers, then the sum is odd.

• 2+5=7

• 3+2=5

• 1+8=9

• 2+4=6

Inductive Reasoning example

1. The flamingos here are all pink. ----> SPECIFIC STATEMENT

2. All flamingos I’ve ever seen are pink.

3. All flamingos must be pink. -----> GENERAL THEORY/IDEA

Inductive Reasoning example

1. It's July, and I saw fireflies in my back yard tonight. (SPECIFIC STATEMENT)

2. I see fireflies in my yard every summer.

3. Fireflies always appear during the summer. (GENERAL THEORY/IDEA)

Multiple Choice

What comes next?

• The next term will have 16 blocks
• The next term will have 15 blocks.
• The next term will have 14 blocks.
• This is the end of the pattern.

​Not p

​And

​Or

Your turn to practice!
Are you ready?

Multiple Choice

What symbol represents a conjunction?

• $$\vee$$  

• $$\wedge$$  

• $$\perp$$  

• $$\sim$$  

Multiple Choice

What symbol represents a disjunction?

• $$\vee$$

• $$\wedge$$

• $$\perp$$

• $$\cong$$

Multiple Choice

Two statements connected by the word "and" is

• A disjunction

• A conjunction

Multiple Choice

A conjunction is true only if

• Both statements are false

• Both statements are true

• One statement is true

• One statement is false

Multiple Choice

A disjunction is true if

• Both statements are false

• Only one statement is true

• At least one statement is true

Multiple Choice

What type of logic is this table showing:

• AND

• OR

• NOR

• XOR

Multiple Choice

Negate the statement:

"I love Geometry"

• Geometry I love

• If I love Geometry, then I do not love Geometry

• If and only if I love Geometry

• I do not love Geometry

Multiple Choice

This symbol, ∨, means?

• and
• or
• not
• implies

Multiple Choice

Determine the truth values of the following compound statements.

$$7$$  is a factor of $$21$$   and $$84$$  

• True

• False

Multiple Choice

Determine the truth values of the following compound statements.

February or October has $$30$$   days.

• True

• False

Multiple Choice

Determine the truth values of the following compound statements.

$$15+7<27$$  and  $$2×8=28$$  

• True

• False

Multiple Choice

Determine the truth values of the following compound statements.

$$3$$  or  $$13$$   is a factor of $$63$$  

• True

• False

Multiple Choice

Determine the truth values of the following compound statements.

$$1.54$$  is a positive number or  $$2^0=0$$  

• True

• False

Multiple Choice

Determine the truth values of the following compound statements.

$$27$$  is a multiple of $$7$$  or  $$\left(-2\right)^2<0$$  

• True

• False

Multiple Choice

Determine the truth values of the following compound statements.

$$49$$  is a perfect square or a prime number

• True

• False

Multiple Choice

Determine the truth values of the following compound statements.

$$2$$  minutes = $$120$$  seconds and $$2$$  hours = $$12$$  minutes

• True

• False

Multiple Choice

Let p and q represent the following simple statements;

p: Frank plays football.

q: Jamir runs track.

r: Edwin plays soccer.

Write each compound statements in symbolic form.

Frank plays football and Jamir runs track.

• p→q

• p↔q

• p Λ q

• p v q

Multiple Choice

Let p and q represent the following simple statements;

p: Frank plays football.

q: Jamir runs track.

r: Edwin plays soccer.

Write each compound statements in symbolic form.

Edwin plays soccer or Frank does not play football.

• $$r\longrightarrow\sim p$$

• $$r\vee\sim p$$

• $$r\wedge\sim p$$

• $$r\longleftrightarrow\sim p$$

Multiple Choice

Used to prove that a conjecture is false.

• Counterexample
• Inductive Reasoning
• Concluding statement
• Conjecture

Multiple Choice

What is the symbol for NOT (negation)?

• ∕
• ∨
• ~
• ∧

Multiple Choice

What are the next 2 numbers of this pattern?
23, 27, 31, 35, ___, ___

• 37, 39
• 40, 45
• 39, 43
• 38, 41

Multiple Choice

What symbol represents a conjunction?

• $$\vee$$  

• $$\wedge$$  

• $$\perp$$  

• $$\sim$$  

Multiple Choice

What symbol represents a disjunction?

• $$\vee$$

• $$\wedge$$

• $$\perp$$

• $$\cong$$

Multiple Choice

Two statements connected by the word "and" is

• A disjunction

• A conjunction

Multiple Choice

A conjunction is true only if

• Both statements are false

• Both statements are true

• One statement is true

• One statement is false

Multiple Choice

A disjunction is true if

• Both statements are false

• Only one statement is true

• At least one statement is true

Multiple Choice

Determine the truth values of the following compound statements.

February or October has $$30$$   days.

• True

• False

Multiple Choice

What are the next 2 numbers of this pattern?
23, 27, 31, 35, ___, ___

• 37, 39
• 40, 45
• 39, 43
• 38, 41

Multiple Choice

Let p represent "Daniel is angry", and let q represent "Daniel is not having fun."
Translate the following symbolic form into written form... 
      p ^ ~ q

• Daniel is not having fun and Daniel is angry.
• Daniel is having fun and Daniel is not angry.
• Daniel is angry and Daniel is having fun. 
• Daniel is angry and Daniel is not having fun.

Multiple Choice

p ∧ q means?

• p and q

• p or q

• p if q

• not p or q

Multiple Choice

Which symbol represents negation?

• $$\ne$$

• $$\infty$$

• $$\sim$$

• $$\wedge$$

Multiple Choice

What type of logic is this table showing:

• AND

• OR

• IF, THEN

Multiple Choice

This symbol, ∧, means?

• and
• or 
• not 
• implies

Multiple Choice

p ∧ q means?

• p and q

• p or q

• p if q

• not p or q

Multiple Choice

Use the logic condition to find the value of X:

A and not B

• True

• False

Multiple Choice

What are the truth values for this table?

~q^~p

• T,T,F,F

• F,F,F,T

• T,F,T,F

• F,F,T,T

Multiple Choice

What are the truth values for this table?

p^q

• F,F,F,F

• T,F,F,T

• T,T,T,T

• T,F,F,F

Multiple Choice

This symbol, ∨, means?

• and
• or
• not
• implies

Multiple Choice

p ∨ q means?

• p and q

• p or q

• p if only q

• neither p or q

Multiple Choice

What are the truth values for this table?

pvq

• T,F,F,F

• T,T,T,T

• T,T,T,F

• F,F,F,F

Multiple Choice

What are the truth values for this table?

pv~q

• T,F,T,T

• F,F,F,T

• T,T,F,T

• T,T,T,T

Multiple Choice

Use the logic condition to find the value of X:

Not A or B

• True

• False

Multiple Choice

What are the truth values for ~p ∨ q

• T F T T

• T T F T

• T T T F

• F T T T

Multiple Choice

What type of logic is this table showing:

• AND

• OR

Multiple Choice

Which of the following is NOT an acceptable set of possible True and False Combinations

Multiple Choice

Fill in the truth table.

Objectives

• To recognize conditional statements and their parts

• To write converses, inverses, and contrapositives.

• A conditional is an if-then statement.

• The hypothesis is the part p following if.

• The conclusion is the part q following then.

Conditional Statements

Logic and Conditional Statements

Multiple Choice

Rewrite the following conditional statement in if-then form: "All numbers divisible by 4 are also divisible by 2"

• If a number is divisible by 4 then it is divisible by 2

• If a number is divisible by 2 then it is divisible by 4

• If a number is not divisible by 4 then it is divisible by 2

• If there is a 4 then there is a 2

Multiple Choice

What symbolic form of a conditional statement mean 'If $$p$$  then $$q$$ ?

• $$p\rightarrow q$$  

• $$q\rightarrow p$$  

• $$\sim q\rightarrow\sim p$$  

• $$\sim p\rightarrow\sim q$$  

The

If-Then

Statement

Hypotheses and Conclusions

A conditional statement (also called an if-then statement) is a statement with a hypothesis followed by a conclusion.

The conclusion is the result of a hypothesis.

Identify hypotheses and conclusions

If you get good grades, Then you will get into a good college.

The part after the "if": you get good grades - is called a hypotheses and the part after the "then" - you will get into a good college - is called a conclusion. ...

IF hypotheses THEN conclusions

If x = 3, then x² = 9.

Hypothesis: "x = 3"

Conclusion: "x² = 9"

IF hypotheses THEN conclusions

If you do your homework, then you can watch TV.

Hypothesis: "you do your homework"

Conclusion: "you can watch TV"

If-then statements might not always be written in the “if-then” form.

Here are some examples of conditional statements:

Statement 1: If you work overtime, then you’ll be paid time-and-a-half.

Statement 2: I’ll wash the car if the weather is nice.

Statement 3: If 2 divides evenly into x, then x is an even number.

Statement 4: I’ll be a millionaire when I win the lottery.

Statement 5: All equiangular triangles are equilateral.

IF hypotheses THEN conclusions

 I'll bring an umbrella if it rains.

Hypothesis: "It rains."

Conclusion: "I'll bring an umbrella."

IF hypotheses THEN conclusions

If a triangle has three congruent sides, it is an equilateral triangle.

Hypothesis: "a triangle has three congruent sides"

Conclusion: "a triangle is an equilateral triangle."

Multiple Choice

What is the hypothesis in this conditional statement?

If it is my birthday, then I eat birthday cake.

• It's my birthday

• I eat birthday cake.

Multiple Choice

What is the hypothesis in this conditional statement?

If 7y - 5 = 9, then y = 2.

• 7y - 5 = 9

• y = 2

Multiple Choice

What is the conclusion in this conditional statement?

If Leon is serving an apple pie, then he is serving a dessert.

• Leon is serving an apple pie

• Leon is serving dessert

Multiple Choice

What is the hypothesis in this conditional statement?

If Greta wants to eat dessert, then she must finish her dinner.

• Greta wants to eat dessert

• Greta must finish her dinner.

Multiple Choice

What is the conclusion in this conditional statement?

If a quadrilateral has no right angles, then it is not a rectangle.

• a quadrilateral has no right angles

• a quadrilateral is not a rectangle.

Multiple Choice

I'll learn how to drive when I am 16 years old.

"I am 16 years old" is considered the:

• hypothesis

• conclusion