2.2 Notes Part 2

Summary

This lecture focuses on evaluating the truth value of conditional statements in the format 'if p then q'. It explains that a statement is false only when the hypothesis (p) is true and the conclusion (q) is false. Examples are provided to illustrate how to identify true and false statements and give counterexamples. The lecture also covers the concepts of negation, converse, inverse, and contrapositive of a statement, using examples to demonstrate how to write and determine their truth values. The importance of understanding logical equivalence between a conditional statement and its contrapositive, and between the converse and inverse, is emphasized.

Takeaways

• 📌 Conditional statements are evaluated as false only when the hypothesis is true and the conclusion is false.
• 🔍 A single counterexample is sufficient to prove a conditional statement false.
• 🗓️ Example: 'If this month is August, then next month is September' is a true statement.
• 📏 Example: 'If two angles are acute, then they are congruent' is false, as shown by 30 and 45-degree angles.
• 🔢 Example: 'If a number greater than two is prime, then five plus four equals eight' is true because the hypothesis is false (4 is not prime).
• ❌ Negation of a statement (not p) flips the truth value: true becomes false, and false becomes true.
• 🔄 The converse of a statement switches the hypothesis and conclusion.
• 🚫 The inverse of a statement negates both the hypothesis and conclusion.
• 🔄 The contrapositive of a statement is the negation of the converse.
• 🔗 Logically equivalent statements (conditional and contrapositive, converse and inverse) share the same truth value.

Q & A

What is the main focus of the script?

The script focuses on explaining conditional statements, specifically how to determine their truth values, and the relationships between different types of statements such as converse, inverse, and contrapositive.

How is the truth value of a conditional statement determined?

A conditional statement is false only when the hypothesis (p) is true and the conclusion (q) is false. Any other combination of truth values results in a true conditional statement.

What is the significance of finding a counterexample for a conditional statement?

A counterexample is crucial because it provides an instance where the hypothesis is true and the conclusion is false, which is the only scenario where a conditional statement is considered false.

Can you provide an example from the script where a statement is true despite having a false hypothesis and conclusion?

Yes, the script gives the example: 'If a number greater than two is prime, then five plus four equals eight.' Here, the hypothesis (an even number greater than two is prime) is false, and the conclusion (five plus four equals eight) is also false, making the overall statement true.

What is the negation of a statement, and how is it represented?

The negation of a statement is the opposite of the original statement. It is represented by inserting the word 'not' into the statement or using the notation '¬p' where 'p' is the original statement.

How do you find the converse of a conditional statement?

To find the converse of a conditional statement 'if p then q', you flip the hypothesis and conclusion, resulting in 'if q then p'.

What is the inverse of a conditional statement, and how is it formed?

The inverse of a conditional statement involves negating both the hypothesis and conclusion. For 'if p then q', the inverse is 'if not p then not q'.

How is the contrapositive of a conditional statement derived?

The contrapositive is formed by negating both the hypothesis and conclusion and then swapping them. For 'if p then q', the contrapositive is 'if not q then not p'.

Why are the conditional and contrapositive of a statement considered logically equivalent?

The conditional and contrapositive are logically equivalent because they share the same truth value. If the conditional is true, the contrapositive is also true, and vice versa.

What is the relationship between the converse and inverse of a statement?

The converse and inverse of a statement are also logically equivalent, sharing the same truth value. If the converse is true, the inverse is true, and if the converse is false, the inverse is false.

Can you provide an example from the script that illustrates the concept of logically equivalent statements?

Yes, the script uses the example of a cat having four paws. The conditional statement 'If an animal is a cat, then it has four paws' and its contrapositive 'If an animal does not have four paws, then it is not a cat' are both true, making them logically equivalent.