Conditionals & counterexamples
Write the following statement as a conditional
"My family eats turkey on Thanksgiving"
What is:
1. If it is Thanksgiving, then my family eats turkey
2. If my family eats turkey, then it is Thanksgiving
What is the converse of:
"If it is raining, then the ground is wet."
If the ground is wet, then it is raining
"If you study, then you will pass the test." You studied.
Is this statement valid or invalid. If valid, what conclusion can be made?
I will pass the test
"If it snows, then school is cancelled." "If school is cancelled, then I sleep in."
What can you conclude?
If it snows, then I sleep in
What are the next 2 numbers in the pattern: 7, 15, 23, 31,...
Add 8; 39, 47
Write the following statement as a conditional
"An obtuse angle measures greater than 90°"
1. If an angle is obtuse, then its measure is greater than 90°
2. If an angle measure is greater than 90°, then it is obtuse
What is the contrapositive of:
"If you live in Ohio, then you live in the United States."
If you do not live in the US, then you do not live in Ohio
If you shop at a store during its annual clearance sale, then you will get 50% off your purchase. You get 50% off your purchase.
Is this statement valid or invalid. If valid, what conclusion can be made?
Invalid, hypothesis does not match
"If a number is divisible by 6, then it is divisible by 3." "If a number is divisible by 3, then the sum of its digits is divisible by 3."
What can you conclude?
If a number is divisible by 6, then the sum of its digits is divisible by 3
1, 4, 9, 16, 25,... What type of numbers are these?
Perfect Squares
12 = 1; 22 = 4; 32 = 9, ...
Provide a counterexample:
"If a number is divisible by 2, then it is divisible by 4."
varies; 6, 10, 14 (anything 2 can divide but 4 cannot)
If x = 3, then x2 = 9.
Write the converse and state whether it's true or false.
If x2 = 9, then x = 3. False; x = 3 and x = -3
"If a polygon has 3 sides, then it is a triangle." Figure ABC has 3 sides.
Is this statement valid or invalid. If valid, what conclusion can be made?
Valid; The polygon is a triangle
If you exercise regularly, then you are healthy. If you exercise regularly, then you eat well.
What can you conclude?
No conclusion, hypothesis is repeated in both statements
A student notices that 12 = 1, 112 = 121, and 1112 = 12321. Using inductive reasoning, predict what 11112 would be equal to
11112 = 1234321 (increases in the middle by one, then decreases after)
Provide a counterexample
"If a figure has four sides, then it is a square."
Rectangles, rhombus, kite, ect.
Decide if the converse is true, then rewrite as a biconditional
"If an angle measures 90°, then it is a right angle."
1. An angle measures 90° if and only if it is a right angle
2. An angle is a right angle if and only if its measure is 90°
"If a student scores above 90%, they earn an A." Maria earned an A.
Is this statement valid or invalid. If valid, what conclusion can be made?
Invalid, hypothesis does not match
If Timmy went to the store, then it is Friday. If Timmy has groceries, then he went to the store.
What can you conclude?
If Timmy has groceries, then it is Friday
The sum of the first n odd numbers equals n2. Verify for n=4 and predict for n=5
n = 4; 1 + 3 + 5 + 7 = 16 = 42
n = 5; 1 + 3 + 5 + 7 + 9 = 25 = 52
Provide a counterexample
"If two angles are supplementary, then one of them is obtuse."
Both angle measures are 90°
Write the two statements that make up the following biconditional
"A month has exactly 28 days if and only if it is February"
1. If a month is February, then it has exactly 28 days
2. If a month has exactly 28 days, then it is February
"If n is divisible by 6, then n is divisible by 3. n = 18."
Is this statement valid or invalid. If valid, what conclusion can be made?
Valid because 18 is divisible by 6, 18 can then be divided by 3
If n is divisible by 4, then n is even. If n is even, then n is divisible by 2."
Apply the Law of Syllogism and provide an example with n=12
If n is divisible by 4, then n is divisible by 2; 12 is divisible by 4, and by 2
A pattern shows 1 = 13, 3 + 5 = 23, 7 + 9 + 11 = 33 Predict the next equation in the sequence.
13 + 15 + 17 + 19 = 43 (or 64)