2.1 Conditional Statements and 2.2 Logical Reasoning

​2.1 and 2.2 Conditional Statements and Logical Reasoning

Mrs. Lett​

Logic and Conditional Statements

Hypothesis and Conclusion of a Conditional

• Hypothesis follows the 'if'

• Conclusion follows the 'then'

Multiple Choice

What is the hypothesis of the conditional?

If you are and American citizen, then you have the right to vote.

• You are an American citizen.

• You have the right to vote.

• You are not an American citizen.

• You do not have the right to vote.

Multiple Choice

What is the Conclusion of the conditional?

If you are and American citizen, then you have the right to vote.

• You are an American citizen.

• You have the right to vote.

• You are not an American citizen.

• You do not have the right to vote.

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 $$q$$ ?

• $$p\rightarrow q$$  

• $$q\rightarrow p$$  

• $$\sim q\rightarrow\sim p$$  

• $$\sim p\rightarrow\sim q$$  

Counter-Example

• Preserves the Hypothesis but contradicts the Conclusion

• ​It is an example that contradicts the statement.

Multiple Choice

If false, which is a counterexample?

• It's True

• My clothes are dry

• I jumped in a pool

• It is raining

Multiple Choice

If false, which is a counterexample?

• It's True

• $$\angle E$$ is obtuse

Multiple Choice

If false, which is a counterexample?

• It's True

• 6

• 12

• 8

Negation

Multiple Choice

Which stands for "if not p then q" ?

• $$\sim p\rightarrow q$$

• $$\sim p\rightarrow\sim q$$

• $$p\rightarrow q$$

• $$p\rightarrow\sim q$$

Multiple Choice

What is the Negation of the statement: 'I went to the store' ?

• I did not go to the store

• I went to Shaws

• I did not leave the house

• I went home

Multiple Choice

What is the Negation of the inequality:  $$x\ge6$$ ?

• $$x<6$$  

• $$x\le6$$  
  

• $$x>6$$  
  

• $$x\ge6$$  
  

Converse, Inverse, and Contrapositive

​Biconditional Statement

Multiple Choice

Given the conditional 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 conditional 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 conditional;

"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

What has to happen in order to write a biconditional statement?

• conditional has to be true
• converse of the conditional has to be true
• all the above
• none of the above

Multiple Choice

What is the biconditional that matches the statement: "A right angle is an angle with 90 degrees."

• If an angle has 90 degrees, then it is a right angle.
• If an angle is a right angle, then it has 90 degrees.
• An angle is a right angle if and only if it has 90 degrees.

​Vocabulary

• ​Conjecture: an unproven statement based on observation.

• ​Inductive Reasoning:Uses a pattern to write a conjecture for a general case.

  • ​Inductive Proofs are common in Computer Science

• ​Deductive Reasoning:uses facts, definition, accepted properties, and laws of logic to form a logical argument

Multiple Choice

If something is an angle, then it is acute. What is an appropriate counter-example?

• 70⁰

• 45⁰

• 120⁰

• True. There is no counterexample.

Multiple Choice

Use inductive reasoning to finish the pattern: 400, 200, 100, 50, ____

• -150
• 25
• 12.5
• 0

Multiple Choice

Inductive Reasoning means...

• Guessing

• Testing and observing patterns to make conjectures

• Explaining why

• Adding numbers together

Multiple Choice

Reasoning based on facts, definitions, accepted properties, and the laws of logic is called ____________ reasoning.

• Inductive

• Deductive

​Laws of Logic

• ​Law of Detachment: if the hypothesis of a true conditional statement is true, then the conclusion is true as well.

• ​Law of Syllogism: If a implies b and b implies c, then a implies c.

Multiple Choice

Use the law of syllogism to draw a conclusion from the two given statements:
Statement 1: If you exercise regularly, then you have a healthy body
Statement 2: If you have a healthy body, then you have more energy

• You have more energy
• If you exercise regularly, then you have more energy
• You have a healthy body
• If you do not have a healthy body, then you do not exercise regularly

Multiple Choice

Use the Law of Detachment to draw a conclusion from the two given statements. If not possible, write not possible.
Statement 1: If I study for one hour each day, then I will score well on the exam
Statement 2: I will score well on the exam

•  I study for 1 hour each day.
• I do not study for 1 hour each day
• If I do well on the exam, I studied for 1 hour each day.
• not possible