Question

name
date
period
geometry
unit 2: logic & proof
quiz 2.2: conditionals, converse, inverse, contrapositive
determine if the conjecture below are true or false. if false, give a counterexample.
1.
4.
8.
match each statement with its symbolic notation.

1. conjunction a. ( q \to p )

b. disjunction b. ( p leftrightarrow q )

1. conditional c. ( p vee q )
2. biconditional d. (

eg p \to
eg q )

1. converse e. (

eg q \to
eg p )

1. contrapositive f. ( p wedge q )
2. inverse g. ( p \to q )

use the statements below to write compound statements. then, determine the truth value.
p: atlanta is the capital of georgia.
q: there are 4 feet in a yard.

1. ( p vee q ):

truth value:

1. ( p wedge q ):

truth value:

1. ( p wedge

eg q ):
truth value:

1. (

eg p vee q ):
truth value:
© gina wilson (all things algebra), llc), 2014 - 2018

Explanation:

Step 1: Analyze Statement \( p \) and \( q \)

• \( p \): Atlanta is the capital of Georgia (True, as Atlanta is the capital of Georgia).
• \( q \): There are 4 feet in a yard (False, since 1 yard = 3 feet).

Step 11: Solve \( p \vee q \)

• Step 1: Recall the definition of disjunction (\(\vee\)): \( p \vee q \) is true if at least one of \( p \) or \( q \) is true.
• Step 2: Substitute the truth values: \( p \) is true, \( q \) is false. Since \( p \) is true, \( p \vee q \) is true.
• The compound statement: "Atlanta is the capital of Georgia or there are 4 feet in a yard."

Step 12: Solve \( p \wedge q \)

• Step 1: Recall the definition of conjunction (\(\wedge\)): \( p \wedge q \) is true only if both \( p \) and \( q \) are true.
• Step 2: Substitute the truth values: \( p \) is true, \( q \) is false. Since \( q \) is false, \( p \wedge q \) is false.
• The compound statement: "Atlanta is the capital of Georgia and there are 4 feet in a yard."

Step 13: Solve \( p \wedge

eg q \)

• Step 1: Find \(

eg q\): \(
eg q\) (negation of \( q \)) is "There are not 4 feet in a yard" (which is true, since 1 yard = 3 feet).

• Step 2: Recall the definition of conjunction (\(\wedge\)): \( p \wedge

eg q \) is true only if both \( p \) and \(
eg q\) are true.

• Step 3: Substitute the truth values: \( p \) is true, \(

eg q\) is true. So \( p \wedge
eg q \) is true.

• The compound statement: "Atlanta is the capital of Georgia and there are not 4 feet in a yard."

Step 14: Solve \(

eg p \vee q \)

• Step 1: Find \(

eg p\): \(
eg p\) (negation of \( p \)) is "Atlanta is not the capital of Georgia" (which is false).

• Step 2: Recall the definition of disjunction (\(\vee\)): \(

eg p \vee q \) is true if at least one of \(
eg p\) or \( q \) is true.

• Step 3: Substitute the truth values: \(

eg p\) is false, \( q \) is false. Since both are false, \(
eg p \vee q \) is false.

• The compound statement: "Atlanta is not the capital of Georgia or there are 4 feet in a yard."

Answer:

1. Compound Statement: "Atlanta is the capital of Georgia or there are 4 feet in a yard."; Truth Value: True
2. Compound Statement: "Atlanta is the capital of Georgia and there are 4 feet in a yard."; Truth Value: False
3. Compound Statement: "Atlanta is the capital of Georgia and there are not 4 feet in a yard."; Truth Value: True
4. Compound Statement: "Atlanta is not the capital of Georgia or there are 4 feet in a yard."; Truth Value: False