Question

practice & problem solving
practice
write each statement as a conditional.
see example 1

1. my skin will get wet if i go swimming.

if i go swimming then my skin will get wet

1. a number that is divisible by 6 is divisible by 3.

if a number is divisible by 6 then it is divisible by 3

1. movie tickets are half - price on tuesdays.

if it is tuesday then movie tickets are half - price
find the truth value of each conditional. explain your reasoning or show a counterexample. see examples 2 and 3

1. if a pair of lines is parallel, then they do not intersect.
2. if the product of two numbers is positive, then the numbers are both positive.

write the negation of the hypothesis and the negation of the conclusion for each conditional. see examples 4 and 5

1. if the sum of the interior angle measures of a polygon is 180, then the polygon is a triangle.
2. if one whole number is odd and the other whole number is even, then the sum of the two numbers is odd.

write each related conditional and determine each truth value for the following conditional. see examples 4 and 5
if an angle measures 100, then it is obtuse.

1. converse
2. contrapositive
3. inverse
4. an employee at an animal shelter wrote the true conditional “if 47% of the dogs at the shelter are female, then 53% of the dogs are male.” can he rewrite this as a true biconditional? explain. see example 6

write two conditionals from each biconditional. see example 7

1. a month has exactly 28 days if and only if it is february.
2. two angles are complementary if and only if their measures add up to 90.
3. the area of a square is s² if and only if the perimeter of the square is 4s.

Explanation:

Step1: Recall conditional statement forms

A conditional statement is of the form "if p then q", where p is the hypothesis and q is the conclusion. The converse is "if q then p", the contra - positive is "if not q then not p", and the inverse is "if not p then not q". A biconditional is of the form "p if and only if q" and can be written as two conditionals: "if p then q" and "if q then p".

Step2: Analyze problem 22

The conditional is "If a pair of lines is parallel, then they do not intersect".

• Truth value: True. By the definition of parallel lines in a plane, parallel lines never intersect.

Step3: Analyze problem 23

The conditional is "If the product of two numbers is positive, then the numbers are both positive".

• Truth value: False. Counter - example: If we take two negative numbers, say - 2 and - 3, their product (-2)×(-3)=6 is positive, but the numbers are not positive.

Step4: Analyze problem 26 (converse of "If an angle measures 100, then it is obtuse")

The converse is "If an angle is obtuse, then it measures 100".

• Truth value: False. An obtuse angle is defined as an angle whose measure is greater than 90 and less than 180. There are many obtuse angles that do not measure 100, like 120, 150 etc.

Step5: Analyze problem 27 (contra - positive of "If an angle measures 100, then it is obtuse")

The contra - positive is "If an angle is not obtuse, then it does not measure 100".

• Truth value: True. If an angle is not obtuse (i.e., it is acute (less than 90) or a right - angle (equal to 90) or a reflex angle (greater than 180)), then it cannot measure 100.

Step6: Analyze problem 28 (inverse of "If an angle measures 100, then it is obtuse")

The inverse is "If an angle does not measure 100, then it is not obtuse".

• Truth value: False. As mentioned before, there are many obtuse angles that do not measure 100.

Step7: Analyze problem 29

The conditional is "If 47% of the dogs at the shelter are female, then 53% of the dogs are male".

• Biconditional: No. A biconditional requires that both the conditional and its converse are true. The converse "If 53% of the dogs are male, then 47% of the dogs are female" is true in the context of only male and female dogs at the shelter. But we are not given that there are only male and female dogs (there could be other genders or non - binary designations in a more inclusive context, or errors in counting etc.). So we cannot write it as a biconditional.

Step8: Analyze problem 30

The biconditional is "A month has exactly 28 days if and only if it is February".

• Two conditionals:
• "If a month has exactly 28 days, then it is February".
• "If it is February, then the month has exactly 28 days" (ignoring leap years for simplicity as the statement doesn't specify).

Step9: Analyze problem 31

The biconditional is "Two angles are complementary if and only if their measures add up to 90".

• Two conditionals:
• "If two angles are complementary, then their measures add up to 90".
• "If the measures of two angles add up to 90, then the two angles are complementary".

Step10: Analyze problem 32

The biconditional is "The area of a square is s² if and only if the perimeter of the square is 4s".

• Two conditionals:
• "If the area of a square is s², then the perimeter of the square is 4s". Let the side length of the square be s. Area A = s² and perimeter P=4s.
• "If the perimeter of the square is 4s, then the area of the square is s²". If P = 4s, then the side length is s, and the area A=s².

Answer:

1. True. Definition of parallel lines.
2. False. Counter - example: (-2)×(-3) = 6.
3. False. There are other obtuse angles.
4. True. Non - obtuse angles can't measure 100.
5. False. There are other obtuse angles.
6. No. Lack of information about only two genders of dogs.
7. If a month has exactly 28 days, then it is February; If it is February, then the month has exactly 28 days.
8. If two angles are complementary, then their measures add up to 90; If the measures of two angles add up to 90, then the two angles are complementary.
9. If the area of a square is s², then the perimeter of the square is 4s; If the perimeter of the square is 4s, then the area of the square is s².