This document defines key vocabulary terms related to statements, conditionals, and biconditionals. It defines statements, truth values, negation, and compound statements. It then defines conjunctions, disjunctions, if-then statements, hypotheses, conclusions, related conditionals, converses, inverses, contrapositives, and biconditionals. It provides examples of writing compound statements and determining their truth values. It also provides examples of identifying hypotheses and conclusions of conditional statements and writing them in if-then form. The document aims to build foundational understanding of propositional logic.

1. Section 2-2
Statements, Conditionals, and Biconditionals

2. Essential Questions
• How do you write compound statements
and determine truth values of compound
statements?
• How do you write conditional statements
and determine truth values of conditional
statements?

3. Vocabulary
1. Statement:
2. Truth Value:
3. Negation:
4. Compound Statement:

4. Vocabulary
1. Statement: A sentence that is either true or
false, often represented as statements p and q
2. Truth Value:
3. Negation:
4. Compound Statement:

5. Vocabulary
1. Statement: A sentence that is either true or
false, often represented as statements p and q
2. Truth Value: The statement is either true (T) or
false (F)
3. Negation:
4. Compound Statement:

6. Vocabulary
1. Statement: A sentence that is either true or
false, often represented as statements p and q
2. Truth Value: The statement is either true (T) or
false (F)
3. Negation: A statement with the opposite
meaning and opposite truth value; the negation
of p is ~p
4. Compound Statement:

7. Vocabulary
1. Statement: A sentence that is either true or
false, often represented as statements p and q
2. Truth Value: The statement is either true (T) or
false (F)
3. Negation: A statement with the opposite
meaning and opposite truth value; the negation
of p is ~p
4. Compound Statement: Two or more
statements joined by and or or

8. Vocabulary
5. Conjunction:
6. Disjunction:
7. Conjunction:
8. If-Then Statements:

9. Vocabulary
5. Conjunction: A compound statement using
the word and or the symbol ⋀
6. Disjunction:
7. Conjunction:
8. If-Then Statements:

10. Vocabulary
5. Conjunction: A compound statement using
the word and or the symbol ⋀
6. Disjunction: A compound statement using the
word or or the symbol ⋁
7. Conjunction:
8. If-Then Statements:

11. Vocabulary
5. Conjunction: A compound statement using
the word and or the symbol ⋀
6. Disjunction: A compound statement using the
word or or the symbol ⋁
7. Conjunction: A statement that fits the if-then
form, providing a connection between the two
phrases
8. If-Then Statements:

12. Vocabulary
5. Conjunction: A compound statement using
the word and or the symbol ⋀
6. Disjunction: A compound statement using the
word or or the symbol ⋁
7. Conjunction: A statement that fits the if-then
form, providing a connection between the two
phrases
8. If-Then Statements: Another name for a
conditional statement; in the form of if p, then q

13. Vocabulary
5. Conjunction: A compound statement using
the word and or the symbol ⋀
6. Disjunction: A compound statement using the
word or or the symbol ⋁
7. Conjunction: A statement that fits the if-then
form, providing a connection between the two
phrases
8. If-Then Statements: Another name for a
conditional statement; in the form of if p, then q
p → q

14. Vocabulary
9. Hypothesis:
10: Conclusion:
11. Related Conditionals:
12. Converse:

15. Vocabulary
9. Hypothesis: The phrase that is the if part of
the conditional; p
10: Conclusion:
11. Related Conditionals:
12. Converse:

16. Vocabulary
9. Hypothesis: The phrase that is the if part of
the conditional; p
10: Conclusion: The phrase that is the then part
of the conditional; q
11. Related Conditionals:
12. Converse:

17. Vocabulary
9. Hypothesis: The phrase that is the if part of
the conditional; p
10: Conclusion: The phrase that is the then part
of the conditional; q
11. Related Conditionals: Statements that are
based off of a given conditional statement
12. Converse:

18. Vocabulary
9. Hypothesis: The phrase that is the if part of
the conditional; p
10: Conclusion: The phrase that is the then part
of the conditional; q
11. Related Conditionals: Statements that are
based off of a given conditional statement
12. Converse: A statement that is created by
switching the hypothesis and conclusion of a
conditional

19. Vocabulary
9. Hypothesis: The phrase that is the if part of
the conditional; p
10: Conclusion: The phrase that is the then part
of the conditional; q
11. Related Conditionals: Statements that are
based off of a given conditional statement
12. Converse: A statement that is created by
switching the hypothesis and conclusion of a
conditional q → p

20. Vocabulary
13. Inverse:
14. Contrapositive:
15. Logically Equivalent:

21. Vocabulary
13. Inverse: A statement that is created by
negating the hypothesis and conclusion of a
conditional
14. Contrapositive:
15. Logically Equivalent:

22. Vocabulary
13. Inverse: A statement that is created by
negating the hypothesis and conclusion of a
conditional
14. Contrapositive:
~ p →~ q
15. Logically Equivalent:

23. Vocabulary
13. Inverse: A statement that is created by
negating the hypothesis and conclusion of a
conditional
14. Contrapositive: A statement that is created
by switching and negating the hypothesis and
conclusion of a conditional
~ p →~ q
15. Logically Equivalent:

24. Vocabulary
13. Inverse: A statement that is created by
negating the hypothesis and conclusion of a
conditional
14. Contrapositive: A statement that is created
by switching and negating the hypothesis and
conclusion of a conditional
~ p →~ q
~ q →~ p
15. Logically Equivalent:

25. Vocabulary
13. Inverse: A statement that is created by
negating the hypothesis and conclusion of a
conditional
14. Contrapositive: A statement that is created
by switching and negating the hypothesis and
conclusion of a conditional
~ p →~ q
~ q →~ p
15. Logically Equivalent: Statements with the
same truth values

26. Vocabulary
13. Inverse: A statement that is created by
negating the hypothesis and conclusion of a
conditional
14. Contrapositive: A statement that is created
by switching and negating the hypothesis and
conclusion of a conditional
~ p →~ q
~ q →~ p
15. Logically Equivalent: Statements with the
same truth values
A conditional and its contrapositive

27. Vocabulary
13. Inverse: A statement that is created by
negating the hypothesis and conclusion of a
conditional
14. Contrapositive: A statement that is created
by switching and negating the hypothesis and
conclusion of a conditional
~ p →~ q
~ q →~ p
15. Logically Equivalent: Statements with the
same truth values
A conditional and its contrapositive
The converse and inverse of a conditional

28. Vocabulary
16. Biconditional:

29. Vocabulary
16. Biconditional: A true compound statement
that consist of the conditional and its converse

30. Vocabulary
16. Biconditional: A true compound statement
that consist of the conditional and its converse
p ↔ q

31. Vocabulary
16. Biconditional: A true compound statement
that consist of the conditional and its converse
p ↔ q
“p if and only if q,” or “p IFF q”

32. Example 1
Use the following statements to write a
compound statement for each conjunction.
Then find its truth value. Explain your reasoning.
p: One meter is 100 mm
q: November has 30 days
r: A line is defined by two points
a. p and q

33. Example 1
Use the following statements to write a
compound statement for each conjunction.
Then find its truth value. Explain your reasoning.
p: One meter is 100 mm
q: November has 30 days
r: A line is defined by two points
a. p and q
One meter is 100 mm and November has 30 days

34. Example 1
Use the following statements to write a
compound statement for each conjunction.
Then find its truth value. Explain your reasoning.
p: One meter is 100 mm
q: November has 30 days
r: A line is defined by two points
a. p and q
One meter is 100 mm and November has 30 days
q is true, but p is false, so the conjunction is false

35. Example 1
Use the following statements to write a
compound statement for each conjunction.
Then find its truth value. Explain your reasoning.
p: One meter is 100 mm
q: November has 30 days
r: A line is defined by two points
b. ~p ⋀ r

36. Example 1
Use the following statements to write a
compound statement for each conjunction.
Then find its truth value. Explain your reasoning.
p: One meter is 100 mm
q: November has 30 days
r: A line is defined by two points
b. ~p ⋀ r
One meter is not 100 mm and a line is defined by two
points

37. Example 1
Use the following statements to write a
compound statement for each conjunction.
Then find its truth value. Explain your reasoning.
p: One meter is 100 mm
q: November has 30 days
r: A line is defined by two points
b. ~p ⋀ r
One meter is not 100 mm and a line is defined by two
points
Both ~p and r are true, so ~p ⋀ r is true

38. Example 2
Use the following statements to write a
compound statement for each conjunction.
Then find its truth value. Explain your reasoning.
p: is proper notation for “ray AB”
q: Kilometers are metric units
r: 15 is a prime number
a. p or q
AB
! "!!

39. Example 2
Use the following statements to write a
compound statement for each conjunction.
Then find its truth value. Explain your reasoning.
p: is proper notation for “ray AB”
q: Kilometers are metric units
r: 15 is a prime number
a. p or q
AB
! "!!
is proper notation for “ray AB” or kilometers are
metric units
AB
! "!!

40. Example 2
Use the following statements to write a
compound statement for each conjunction.
Then find its truth value. Explain your reasoning.
p: is proper notation for “ray AB”
q: Kilometers are metric units
r: 15 is a prime number
a. p or q
AB
! "!!
Both p and q are true, so p or q is true
is proper notation for “ray AB” or kilometers are
metric units
AB
! "!!

41. Example 2
Use the following statements to write a
compound statement for each conjunction.
Then find its truth value. Explain your reasoning.
p: is proper notation for “ray AB”
q: Kilometers are metric units
r: 15 is a prime number
b. q ⋁ r
AB
! "!!

42. Example 2
Use the following statements to write a
compound statement for each conjunction.
Then find its truth value. Explain your reasoning.
p: is proper notation for “ray AB”
q: Kilometers are metric units
r: 15 is a prime number
b. q ⋁ r
AB
! "!!
Kilometers are metric units or 15 is a prime number

43. Example 2
Use the following statements to write a
compound statement for each conjunction.
Then find its truth value. Explain your reasoning.
p: is proper notation for “ray AB”
q: Kilometers are metric units
r: 15 is a prime number
b. q ⋁ r
AB
! "!!
Since one of the statements (q) is true, q ⋁ r is true
Kilometers are metric units or 15 is a prime number

44. Example 2
Use the following statements to write a
compound statement for each conjunction.
Then find its truth value. Explain your reasoning.
p: is proper notation for “ray AB”
q: Kilometers are metric units
r: 15 is a prime number
c. ~p ⋁ r
AB
! "!!

45. Example 2
Use the following statements to write a
compound statement for each conjunction.
Then find its truth value. Explain your reasoning.
p: is proper notation for “ray AB”
q: Kilometers are metric units
r: 15 is a prime number
c. ~p ⋁ r
AB
! "!!
is not proper notation for “ray AB” or 15 is a
prime number
AB
! "!!

46. Example 2
Use the following statements to write a
compound statement for each conjunction.
Then find its truth value. Explain your reasoning.
p: is proper notation for “ray AB”
q: Kilometers are metric units
r: 15 is a prime number
c. ~p ⋁ r
AB
! "!!
Since both ~p and r are false, ~p ⋁ r is false
is not proper notation for “ray AB” or 15 is a
prime number
AB
! "!!

47. Example 3
Identify the hypothesis and conclusion of each
statement.
a. If a polygon has eight sides, then it is an octagon.
b. Matt Mitarnowski will advance to the next level if he
completes the Towers of Hanoi in his computer game.

48. Example 3
Identify the hypothesis and conclusion of each
statement.
a. If a polygon has eight sides, then it is an octagon.
b. Matt Mitarnowski will advance to the next level if he
completes the Towers of Hanoi in his computer game.

49. Example 3
Identify the hypothesis and conclusion of each
statement.
a. If a polygon has eight sides, then it is an octagon.
Hypothesis
b. Matt Mitarnowski will advance to the next level if he
completes the Towers of Hanoi in his computer game.

50. Example 3
Identify the hypothesis and conclusion of each
statement.
a. If a polygon has eight sides, then it is an octagon.
Hypothesis
b. Matt Mitarnowski will advance to the next level if he
completes the Towers of Hanoi in his computer game.

51. Example 3
Identify the hypothesis and conclusion of each
statement.
a. If a polygon has eight sides, then it is an octagon.
Hypothesis Conclusion
b. Matt Mitarnowski will advance to the next level if he
completes the Towers of Hanoi in his computer game.

52. Example 3
Identify the hypothesis and conclusion of each
statement.
a. If a polygon has eight sides, then it is an octagon.
Hypothesis Conclusion
b. Matt Mitarnowski will advance to the next level if he
completes the Towers of Hanoi in his computer game.
Hypothesis

53. Example 3
Identify the hypothesis and conclusion of each
statement.
a. If a polygon has eight sides, then it is an octagon.
Hypothesis Conclusion
b. Matt Mitarnowski will advance to the next level if he
completes the Towers of Hanoi in his computer game.
Hypothesis
Conclusion

54. Example 4
Identify the hypothesis and conclusion of each
statement. Then write each statement in the if-then
form.
a. Measured distance is positive.

55. Example 4
Identify the hypothesis and conclusion of each
statement. Then write each statement in the if-then
form.
a. Measured distance is positive.
Hypothesis: A distance is measured

56. Example 4
Identify the hypothesis and conclusion of each
statement. Then write each statement in the if-then
form.
a. Measured distance is positive.
Hypothesis: A distance is measured
Conclusion: It is positive

57. Example 4
Identify the hypothesis and conclusion of each
statement. Then write each statement in the if-then
form.
a. Measured distance is positive.
Hypothesis: A distance is measured
Conclusion: It is positive
If a distance is measured, then it is positive.

58. Example 4
Identify the hypothesis and conclusion of each
statement. Then write each statement in the if-then
form.
b. A six-sided polygon is a hexagon

59. Example 4
Identify the hypothesis and conclusion of each
statement. Then write each statement in the if-then
form.
b. A six-sided polygon is a hexagon
Hypothesis: A polygon has six sides

60. Example 4
Identify the hypothesis and conclusion of each
statement. Then write each statement in the if-then
form.
b. A six-sided polygon is a hexagon
Hypothesis: A polygon has six sides
Conclusion: It is a hexagon

61. Example 4
Identify the hypothesis and conclusion of each
statement. Then write each statement in the if-then
form.
b. A six-sided polygon is a hexagon
Hypothesis: A polygon has six sides
Conclusion: It is a hexagon
If a polygon has six sides, then it is a hexagon.

62. Example 5
Determine the truth value of each conditional
statement. If true, explain your reasoning. If false,
give a counter example.
a. If you subtract a whole number from another whole
number, the result is also a whole number.

63. Example 5
Determine the truth value of each conditional
statement. If true, explain your reasoning. If false,
give a counter example.
a. If you subtract a whole number from another whole
number, the result is also a whole number.
False

64. Example 5
Determine the truth value of each conditional
statement. If true, explain your reasoning. If false,
give a counter example.
a. If you subtract a whole number from another whole
number, the result is also a whole number.
False
5 − 11 = −6

65. Example 5
Determine the truth value of each conditional
statement. If true, explain your reasoning. If false,
give a counter example.
b. If last month was September, then this month is
October.

66. Example 5
Determine the truth value of each conditional
statement. If true, explain your reasoning. If false,
give a counter example.
b. If last month was September, then this month is
October.
True

67. Example 5
Determine the truth value of each conditional
statement. If true, explain your reasoning. If false,
give a counter example.
c. When a rectangle has an obtuse angle, it is a
parallelogram.

68. Example 5
Determine the truth value of each conditional
statement. If true, explain your reasoning. If false,
give a counter example.
c. When a rectangle has an obtuse angle, it is a
parallelogram.
A rectangle cannot have an obtuse angle, so we
cannot test this. All rectangles are
parallelograms.

69. Example 5
Determine the truth value of each conditional
statement. If true, explain your reasoning. If false,
give a counter example.
c. When a rectangle has an obtuse angle, it is a
parallelogram.
True: False Hypothesis
A rectangle cannot have an obtuse angle, so we
cannot test this. All rectangles are
parallelograms.

70. Example 6
Determine the converse, inverse, and contrapositive
for the following statement. Then determine if the
new statement is true. If false, give a
counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Converse:

71. Example 6
Determine the converse, inverse, and contrapositive
for the following statement. Then determine if the
new statement is true. If false, give a
counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Converse:
If MN ≅ NO, then N is the midpoint of MO.

72. Example 6
Determine the converse, inverse, and contrapositive
for the following statement. Then determine if the
new statement is true. If false, give a
counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Converse:
If MN ≅ NO, then N is the midpoint of MO.
False

73. Example 6
Determine the converse, inverse, and contrapositive
for the following statement. Then determine if the
new statement is true. If false, give a
counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Converse:
If MN ≅ NO, then N is the midpoint of MO.
False
M, N, and O might not be collinear

74. Example 6
Determine the converse, inverse, and contrapositive
for the following statement. Then determine if the
new statement is true. If false, give a
counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Converse:
If MN ≅ NO, then N is the midpoint of MO.
False
M, N, and O might not be collinear
M N
O

75. Example 6
Determine the converse, inverse, and contrapositive
for the following statement. Then determine if the
new statement is true. If false, give a
counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Inverse:

76. Example 6
Determine the converse, inverse, and contrapositive
for the following statement. Then determine if the
new statement is true. If false, give a
counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Inverse:
If N is not the midpoint of MO, then MN ≅ NO.

77. Example 6
Determine the converse, inverse, and contrapositive
for the following statement. Then determine if the
new statement is true. If false, give a
counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Inverse:
False
If N is not the midpoint of MO, then MN ≅ NO.

78. Example 6
Determine the converse, inverse, and contrapositive
for the following statement. Then determine if the
new statement is true. If false, give a
counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Inverse:
False
If N is not the midpoint of MO, then MN ≅ NO.
If N is not on MO, then MN could be
congruent to NO.

79. Example 6
Determine the converse, inverse, and contrapositive
for the following statement. Then determine if the
new statement is true. If false, give a
counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Inverse:
False
If N is not the midpoint of MO, then MN ≅ NO.
If N is not on MO, then MN could be
congruent to NO. M N O

80. Example 6
Determine the converse, inverse, and contrapositive
for the following statement. Then determine if the
new statement is true. If false, give a
counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Contrapositive:

81. Example 6
Determine the converse, inverse, and contrapositive
for the following statement. Then determine if the
new statement is true. If false, give a
counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Contrapositive:
If MN ≅ NO, then N is not the midpoint of MO.

82. Example 6
Determine the converse, inverse, and contrapositive
for the following statement. Then determine if the
new statement is true. If false, give a
counterexample.
If N is the midpoint of MO, then MN ≅ NO.
Contrapositive:
True
If MN ≅ NO, then N is not the midpoint of MO.