2 2 problem solving conditional statements

Geometry: Common Core (15th Edition)

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HGeo 2.2 Conditional Statements.ppt

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• 1. Holt McDougal Geometry 2-2 Conditional Statements Students will… Identify, write, and analyze the truth value of conditional statements. Write the inverse, converse, and contrapositive of a conditional statement. Objectives
• 2. Holt McDougal Geometry 2-2 Conditional Statements Warm Up Determine if each statement is true or false. 1. The measure of an obtuse angle is less than 90°. 2. All perfect-square numbers are positive. 3. Every prime number is odd. 4. Any three points are coplanar. F T F T
• 3. Holt McDougal Geometry 2-2 Conditional Statements conditional statement hypothesis conclusion truth value negation converse inverse contrapostive logically equivalent statements Vocabulary
• 4. Holt McDougal Geometry 2-2 Conditional Statements By phrasing a conjecture as an if-then statement, you can quickly identify its hypothesis and conclusion.
• 5. Holt McDougal Geometry 2-2 Conditional Statements Identify the hypothesis and conclusion of each conditional. Example 1: Identifying the Parts of a Conditional Statement A. If today is Thanksgiving Day, then today is Thursday. B. A number is a rational number if it is an integer. Hypothesis: Today is Thanksgiving Day. Conclusion: Today is Thursday. Hypothesis: A number is an integer. Conclusion: The number is a rational number.
• 6. Holt McDougal Geometry 2-2 Conditional Statements “If p, then q” can also be written as “if p, q,” “q, if p,” “p implies q,” “q when p,” “q because p,” and “p only if q.” Writing Math C. I sneeze, when I have a cold. Hypothesis: I have a cold. Conclusion: I sneeze. D. It snows because it is cold. Hypothesis: It is cold. Conclusion: It snows.
• 7. Holt McDougal Geometry 2-2 Conditional Statements *Sentences without "if" and "then" can be written as conditionals. *Identify the hypothesis and conclusion by figuring out which part of the statement depends on the other.
• 8. Holt McDougal Geometry 2-2 Conditional Statements Write a conditional statement from the following. Example 2A: Writing a Conditional Statement An obtuse triangle has exactly one obtuse angle. If a triangle is obtuse, then it has exactly one obtuse angle. Identify the hypothesis and the conclusion. An obtuse triangle has exactly one obtuse angle.
• 9. Holt McDougal Geometry 2-2 Conditional Statements Write a conditional statement from the following. Example 2B: Writing a Conditional Statement If an animal is a blue jay, then it is a bird. The inner oval represents the hypothesis, and the outer oval represents the conclusion.
• 10. Holt McDougal Geometry 2-2 Conditional Statements A conditional statement has a truth value of either true (T) or false (F). It is false only when the hypothesis is true and the conclusion is false. To show that a conditional statement is false, you need to find only one counterexample where the hypothesis is true and the conclusion is false.
• 11. Holt McDougal Geometry 2-2 Conditional Statements Example 3A Determine if the conditional “If a number is odd, then it is divisible by 3” is true. If false, give a counterexample. An example of an odd number is 7. It is not divisible by 3. In this case, the hypothesis is true, but the conclusion is false. Since you can find a counterexample, the conditional is false.
• 12. Holt McDougal Geometry 2-2 Conditional Statements Determine if the conditional is true. If false, give a counterexample. Example 3B: Analyzing the Truth Value of a Conditional Statement You can have acute angles with measures of 80° and 30°. In this case, the hypothesis is true, but the conclusion is false. If two angles are acute, then they are congruent. Since you can find a counterexample, the conditional is false.
• 13. Holt McDougal Geometry 2-2 Conditional Statements Determine if the conditional is true. If false, give a counterexample. Example 3C: Analyzing the Truth Value of a Conditional Statement An even number greater than 2 will never be prime, so the hypothesis is false. 5 + 4 is not equal to 8, so the conclusion is false. However, the conditional is true because the hypothesis is false. If an even number greater than 2 is prime, then 5 + 4 = 8.
• 14. Holt McDougal Geometry 2-2 Conditional Statements *The negation of statement p is “not p,” written as ~p. *The negation of a true statement is false, and the negation of a false statement is true.
• 15. Holt McDougal Geometry 2-2 Conditional Statements Definition Symbols A conditional is a statement that can be written in the form "If p, then q." p  q Related Conditionals
• 16. Holt McDougal Geometry 2-2 Conditional Statements Definition Symbols The converse is the statement formed by exchanging the hypothesis and conclusion. q  p Related Conditionals
• 17. Holt McDougal Geometry 2-2 Conditional Statements Definition Symbols The inverse is the statement formed by negating the hypothesis and conclusion. ~p  ~q Related Conditionals
• 18. Holt McDougal Geometry 2-2 Conditional Statements Definition Symbols The contrapositive is the statement formed by both exchanging and negating the hypothesis and conclusion. ~q  ~p Related Conditionals
• 19. Holt McDougal Geometry 2-2 Conditional Statements Write the converse, inverse, and contrapostive of the conditional statement “If an animal is a cat, then it has four paws.” Find the truth value of each. Check It Out! Example 4 If an animal is a cat, then it has four paws.
• 20. Holt McDougal Geometry 2-2 Conditional Statements Check It Out! Example 4 Inverse: If an animal is not a cat, then it does not have 4 paws. Converse: If an animal has 4 paws, then it is a cat. Contrapositive: If an animal does not have 4 paws, then it is not a cat; True. If an animal is a cat, then it has four paws. There are other animals that have 4 paws that are not cats, so the converse is false. There are animals that are not cats that have 4 paws, so the inverse is false. Cats have 4 paws, so the contrapositive is true.
• 21. Holt McDougal Geometry 2-2 Conditional Statements *Related conditional statements that have the same truth value are called logically equivalent statements. A conditional and its contrapositive are logically equivalent, and so are the converse and inverse.
• 22. Holt McDougal Geometry 2-2 Conditional Statements The logical equivalence of a conditional and its contrapositive is known as the Law of Contrapositive. Helpful Hint
• 23. Holt McDougal Geometry 2-2 Conditional Statements Lesson Quiz: Part I Identify the hypothesis and conclusion of each conditional. 1. A triangle with one right angle is a right triangle. 2. All even numbers are divisible by 2. 3. Determine if the statement “If n2 = 144, then n = 12” is true. If false, give a counterexample. H: A number is even. C: The number is divisible by 2. H: A triangle has one right angle. C: The triangle is a right triangle. False; n = –12.
• 24. Holt McDougal Geometry 2-2 Conditional Statements Lesson Quiz: Part II Identify the hypothesis and conclusion of each conditional. 4. Write the converse, inverse, and contrapositive of the conditional statement “If Maria’s birthday is February 29, then she was born in a leap year.” Find the truth value of each. Converse: If Maria was born in a leap year, then her birthday is February 29; False. Inverse: If Maria’s birthday is not February 29, then she was not born in a leap year; False. Contrapositive: If Maria was not born in a leap year, then her birthday is not February 29; True.
• 25. Holt McDougal Geometry 2-2 Conditional Statements Check It Out! Example 2 Write a conditional statement from the sentence “Two angles that are complementary are acute.” If two angles are complementary, then they are acute. Identify the hypothesis and the conclusion. Two angles that are complementary are acute.
• 26. Holt McDougal Geometry 2-2 Conditional Statements Check It Out! Example 1 "A number is divisible by 3 if it is divisible by 6." Identify the hypothesis and conclusion of the statement. Hypothesis: A number is divisible by 6. Conclusion: A number is divisible by 3.
• 27. Holt McDougal Geometry 2-2 Conditional Statements Determine if the conditional is true. If false, give a counterexample. Example 3A: Analyzing the Truth Value of a Conditional Statement If this month is August, then next month is September. When the hypothesis is true, the conclusion is also true because September follows August. So the conditional is true.
• 28. Holt McDougal Geometry 2-2 Conditional Statements Write the converse, inverse, and contrapositive of the conditional statement. Use the Science Fact to find the truth value of each. Example 4: Biology Application If an animal is an adult insect, then it has six legs.
• 29. Holt McDougal Geometry 2-2 Conditional Statements Example 4: Biology Application Inverse: If an animal is not an adult insect, then it does not have six legs. Converse: If an animal has six legs, then it is an adult insect. If an animal is an adult insect, then it has six legs. No other animals have six legs so the converse is true. Contrapositive: If an animal does not have six legs, then it is not an adult insect. Adult insects must have six legs. So the contrapositive is true. No other animals have six legs so the converse is true.

Conditional Statement – Definition, Truth Table, Examples, FAQs

What is a conditional statement, how to write a conditional statement, what is a biconditional statement, solved examples on conditional statements, practice problems on conditional statements, frequently asked questions about conditional statements.

A conditional statement is a statement that is written in the “If p, then q” format. Here, the statement p is called the hypothesis and q is called the conclusion. It is a fundamental concept in logic and mathematics. 

Conditional statement symbol :  p → q

A conditional statement consists of two parts.

• The “if” clause, which presents a condition or hypothesis.
• The “then” clause, which indicates the consequence or result that follows if the condition is true. 

Example : If you brush your teeth, then you won’t get cavities.

Hypothesis (Condition): If you brush your teeth

Conclusion (Consequence): then you won’t get cavities 

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Conditional Statement: Definition

A conditional statement is characterized by the presence of “if” as an antecedent and “then” as a consequent. A conditional statement, also known as an “if-then” statement consists of two parts:

• The “if” clause (hypothesis): This part presents a condition, situation, or assertion. It is the initial condition that is being considered.
• The “then” clause (conclusion): This part indicates the consequence, result, or action that will occur if the condition presented in the “if” clause is true or satisfied. 

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Representation of Conditional Statement

The conditional statement of the form ‘If p, then q” is represented as p → q. 

It is pronounced as “p implies q.”

Different ways to express a conditional statement are:

• p implies q
• p is sufficient for q
• q is necessary for p

Parts of a Conditional Statement

There are two parts of conditional statements, hypothesis and conclusion. The hypothesis or condition will begin with the “if” part, and the conclusion or action will begin with the “then” part. A conditional statement is also called “implication.”

Conditional Statements Examples:

Example 1: If it is Sunday, then you can go to play. 

Hypothesis: If it is Sunday

Conclusion: then you can go to play. 

Example 2: If you eat all vegetables, then you can have the dessert.

Condition: If you eat all vegetables

Conclusion: then you can have the dessert 

To form a conditional statement, follow these concise steps:

Step 1 : Identify the condition (antecedent or “if” part) and the consequence (consequent or “then” part) of the statement.

Step 2 : Use the “if… then…” structure to connect the condition and consequence.

Step 3 : Ensure the statement expresses a logical relationship where the condition leads to the consequence.

Example 1 : “If you study (condition), then you will pass the exam (consequence).” 

This conditional statement asserts that studying leads to passing the exam. If you study (condition is true), then you will pass the exam (consequence is also true).

Example 2 : If you arrange the numbers from smallest to largest, then you will have an ascending order.

Hypothesis: If you arrange the numbers from smallest to largest

Conclusion: then you will have an ascending order

Truth Table for Conditional Statement

The truth table for a conditional statement is a table used in logic to explore the relationship between the truth values of two statements. It lists all possible combinations of truth values for “p” and “q” and determines whether the conditional statement is true or false for each combination. 

The truth value of p → q is false only when p is true and q is False. 

If the condition is false, the consequence doesn’t affect the truth of the conditional; it’s always true.

In all the other cases, it is true.

The truth table is helpful in the analysis of possible combinations of truth values for hypothesis or condition and conclusion or action. It is useful to understand the presence of truth or false statements. 

Converse, Inverse, and Contrapositive

The converse, inverse, and contrapositive are three related conditional statements that are derived from an original conditional statement “p → q.” 

Consider a conditional statement: If I run, then I feel great.

• Converse: 

The converse of “p → q” is “q → p.” It reverses the order of the original statement. While the original statement says “if p, then q,” the converse says “if q, then p.” 

Converse: If I feel great, then I run.

• Inverse: 

The inverse of “p → q” is “~p → ~q,” where “” denotes negation (opposite). It negates both the antecedent (p) and the consequent (q). So, if the original statement says “if p, then q,” the inverse says “if not p, then not q.”

Inverse : If I don’t run, then I don’t feel great.

• Contrapositive: 

The contrapositive of “p → q” is “~q → ~p.” It reverses the order and also negates both the statements. So, if the original statement says “if p, then q,” the contrapositive says “if not q, then not p.”

Contrapositive: If I don’t feel great, then I don’t run.

A biconditional statement is a type of compound statement in logic that expresses a bidirectional or two-way relationship between two statements. It asserts that “p” is true if and only if “q” is true, and vice versa. In symbolic notation, a biconditional statement is represented as “p ⟺ q.”

In simpler terms, a biconditional statement means that the truth of “p” and “q” are interdependent. 

If “p” is true, then “q” must also be true, and if “q” is true, then “p” must be true. Conversely, if “p” is false, then “q” must be false, and if “q” is false, then “p” must be false. 

Biconditional statements are often used to express equality, equivalence, or conditions where two statements are mutually dependent for their truth values. 

Examples : 

• I will stop my bike if and only if the traffic light is red.  
• I will stay if and only if you play my favorite song.

Facts about Conditional Statements

• The negation of a conditional statement “p → q” is expressed as “p and not q.” It is denoted as “𝑝 ∧ ∼𝑞.” 
• The conditional statement is not logically equivalent to its converse and inverse.
• The conditional statement is logically equivalent to its contrapositive. 
• Thus, we can write p → q ∼q → ∼p

In this article, we learned about the fundamentals of conditional statements in mathematical logic, including their structure, parts, truth tables, conditional logic examples, and various related concepts. Understanding conditional statements is key to logical reasoning and problem-solving. Now, let’s solve a few examples and practice MCQs for better comprehension.

Example 1: Identify the hypothesis and conclusion. 

If you sing, then I will dance.

Solution : 

Given statement: If you sing, then I will dance.

Here, the antecedent or the hypothesis is “if you sing.”

The conclusion is “then I will dance.”

Example 2: State the converse of the statement: “If the switch is off, then the machine won’t work.” 

Here, p: The switch is off

q: The machine won’t work.

The conditional statement can be denoted as p → q.

Converse of p → q is written by reversing the order of p and q in the original statement.

Converse of  p → q is q → p.

Converse of  p → q: q → p: If the machine won’t work, then the switch is off.

Example 3: What is the truth value of the given conditional statement? 

If 2+2=5 , then pigs can fly.

Solution:  

q: Pigs can fly.

The statement p is false. Now regardless of the truth value of statement q, the overall statement will be true. 

F → F = T

Hence, the truth value of the statement is true. 

Conditional Statement - Definition, Truth Table, Examples, FAQs

Attend this quiz & Test your knowledge.

What is the antecedent in the given conditional statement? If it’s sunny, then I’ll go to the beach.

A conditional statement can be expressed as, what is the converse of “a → b”, when the antecedent is true and the consequent is false, the conditional statement is.

What is the meaning of conditional statements?

Conditional statements, also known as “if-then” statements, express a cause-and-effect or logical relationship between two propositions.

When does the truth value of a conditional statement is F?

A conditional statement is considered false when the antecedent is true and the consequent is false.

What is the contrapositive of a conditional statement?

The contrapositive reverses the order of the statements and also negates both the statements. It is equivalent in truth value to the original statement.

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