GM-U1 Unit Review: Logic

Identification of patterns to establish a rule, or complete a pattern.

Inductive Reason

If, then statements.
If = hypothesis.
Then = conclusion.

​Conditionals

Concepts:

Use of data to draw a conclusion.

Deductive Reason

Statements and reasons used to prove a given statement.

​Proofs

Concepts:

A logical operation that reverses the truth value of a statement.

Negation

Prove a statement using contradiction or contrapositive.

​Indirect Proofs

Concepts:

Identifying Contradictions

Which two statements contradict each other?

1) FG ∥ KL
2) FG ≅ KL
3) FG ⊥ KL

Identifying Contradictions

Which two statements contradict each other?

1) FG ∥ KL
2) FG ≅ KL
3) FG ⊥ KL

• Segments can be ∥ and ≅. Statements I and II do not contradict each other.

• Segments can be ≅ and ⊥. Statements II and III do not contradict each other.

• ∥ segments do not intersect, so they cannot be ⊥. Statements I and III contradict each other.

Multiple Choice

What is the assumption for solving indirect proofs by contradiction?

• Assume the opposite of what you want to prove.

• Assume what you want to prove.

• Assume the given statement.

• Create the contrapositive.

Multiple Choice

What is the conclusion when proving by contradiction?

• Your assumption was correct and now you are done.

• Conclude that the proof is done.

• Contradiction shows that the assumption was false, so what you're proving is false.

• Assumption and given conflict.

Multiple Choice

Indirect Reasoning

• Type of reasoning where you go from one step to the next.

• Type of reasoning where one possibility is proven false.

• Type of reasoning where all possibilities fail.

• A difficult reasoning problem.

Multiple Choice

First step to prove an equilateral triangle does not have a right angle by contradiction:

• Assume the triangle has a right angle.

• Assume the triangle does not have a right angle.

• Assume the triangle has equal sides.

• Assume the triangle does not have any angles.

Multiple Choice

What is a direct proof?

• A proof that always involves the multiplication of two values.

• A proof that can only use number properties to show that a certain

  statement is false.

• A proof that assumes a statement's hypothesis is true and uses a series of

  logical deductions to conclude that the statement's conclusion is true.

• A proof that assumes that the statement being proven is false and then

  attempts to find a contradiction to that assumption proving the original

  statement to be true.

Multiple Choice

Given ∠𝐴 ≅ ∠𝐵 and ∠𝐵 ≅ ∠𝐶. Prove ∠𝐴 ≅ ∠𝐶. What is the reason for the statement

𝑚∠𝐴 = 𝑚∠𝐶 in Step 3 of the proof?

• Addition Property of Equality

• Distributive Property of Equality

• Subtraction Property of Equality

• Transitive Property of Equality

Multiple Choice

What is the justification (reason)?

• Definition of angle bisector.

• Definition of segment bisector.

• Definition of a midpoint.

• Substitution

Multiple Choice

Given that angles A and B form a linear pair what is the next step we can conclude by definition of a linear pair?

• m∠A + m∠B = 90
• A and B are both acute angles
• m∠A + m∠B = 180
• A and B are both obtuse angles

Multiple Choice

If ∠A and ∠B are complementary,
then m∠A + m∠B = 90°.

• Definition of Supplementary Angles
• Angle Addition Postulate
• Definition of Right Angle
• Definition of Complementary Angles

Multiple Choice

If ∠CAT is a right angle, then m∠CAT = 90°.

• Definition of Right Angle
• Definition of Congruent Angles
• Angle Addition Postulate
• Segment Addition Postulate

Multiple Choice

If AC ⊥ BD, then ∠DBC is a right angle.

• Definition of Perpendicular Lines
• Definition of Right Angle
• Angle Addition Postulate
• Segment Addition Postulate

Multiple Choice

If DB bisects ∠ABC, then ∠ABD = ∠CBD.

• Definition of Congruent Angles
• Definition of Angle Bisector
• Angle Addition Postulate
• Definition of Right Angle

Multiple Choice

Given the statement;

"If I am hungry, then I eat."

Which of the following is the inverse?

• If I eat, then I am hungry.

• If I don't eat then I am not hungry

• If I am not hungry, then I don't eat.

• If I am not hungry, then I eat.

Multiple Choice

Given the statement;

"If Max gets good grade, then he won't be grounded"

Which of the following is the contrapositive?

• If Max isn't grounded then he gets good grades.

• If Max doesn't get good grade then he will be grounded.

• If Max gets grounded then he doesn't get good grades.

• If Max gets good grades, the he will get grounded.

Multiple Choice

Given the statement;

"If Jose eats fish, then he has an allergic reaction"

Which of the following is the converse?

• None of these

• If Jose doesn't have an allergic reaction, then he didn't eat fish.

• If Jose doesn't eat fish, then he doesn't have an allergic reaction.

• If Jose has an allergic reaction, then he ate fish.

Multiple Choice

Given the statement;

"If it snows then the school will be closed"

Which is the inverse?

• If the school is open, then it didn't snow.

• If the school is closed, then it snowed.

• If it doesn't snow, then the school will be open.

• If the school is open, then it snowed.

Multiple Choice

Given the statement;

"If it snows then the school will be closed"

Which is the hypothesis?

• Then it didn't snow.

• If the school is closed.

• If it snows.

• Then the school will be closed.

Multiple Choice

Given the statement;

"If it snows then the school will be closed"

Which is the conclusion?

• Then it didn't snow.

• If the school is closed.

• If it snows.

• Then the school will be closed.

Multiple Choice

What is the contrapositive to this?

• If an angle is not greater than 90, then it is not obtuse

• If an angle is not obtuse then it is not greater than 90.

• If an angle is 90, then it is right

• If an angle is not greater than 90 then it must be acute

Multiple Choice

When taking the inverse we _____________ the hypothesis and conclusion.

• negate
• switch
• switch and negate
• leave the same

Multiple Choice

For the contrapositive we _________ the hypothesis and conclusion

• flip

• negate
• flip and negate
• do nothing to

Multiple Choice

Which of the following is the "if-then" form of the sentence:

"All students should study."

• If you are a student, then you should study.

• If you study, then you are a student.

• If a student studies, then they will earn good grades.

• All studying should be done by students.

Multiple Choice

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

• 4
• -4
• 3
• -3

Multiple Choice

How can you prove a conjecture is false?

• Counterexample
• Not Possible
• Patterns

Multiple Choice

Which of the following conjectures is false?

• The product of two even numbers is even.
• The sum of two even numbers is even.
• The product of two odd numbers is odd.
• The sum of two odd numbers is odd.

Multiple Choice

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

• positive
• negative
• odd
• even

Multiple Choice

Find a pattern in the sequence.  Use the pattern to show the next two terms.
1, 3, 7, 15, 31, ___ , ___

• 60, 120
• 63, 127
• 54, 116
• 57, 121

Multiple Choice

Which is a counterexample of: Any number divisible by 2 is divisible by 4.

• 15÷2
• 20÷4
• 16÷4
• 10÷2

Multiple Choice

How many squares are in Figure 4?

• 10

• 11

• 12

• 13

• 14

Multiple Choice

What is the rule?

• Multiply by 2
• Multiply by 3
• Add 3
• Add 4

Multiple Choice

What does deductive reasoning use to form its arguments?

• patterns

• specific cases

• observed examples

• facts, definitions, accepted properties, and laws of logic

Multiple Choice

How would you describe the Law of Detachment symbolically?

• if p -> q is true and q is true; then p is true

• if p -> q is true and p is true; then q is true

• if p -> q and q -> r, then p -> r

Multiple Choice

How would you describe the Law of Syllogism symbolically?

• if p -> q is true and q is true, then p is true.

• if p -> q is true and p is true, then q is true.

• if p -> q is true and q -> r is true, then p -> r is true

Multiple Choice

If the school cafeteria sells pizza today, Greg is going to eat four pieces. Pizza is on sale for school lunch today.

Using deductive reasoning, what statement can you make based on this information?

• Lots of people will buy pizza.

• The school cafeteria sells pizza.

• Greg will eat four pieces of pizza.

• Greg needs to go on a diet.