Pat urns, buleez
What are the next three terms of this sequence?
1, 3, 6, 10, ...
15, 21, 28
Determine the converse, inverse, and contrapositive of the following. Determine the truth value and provide counterexamples if necessary.
If you are a teenager, then you are younger than 20.
Converse: If you are younger than 20, then you are a teenager. F (think 7 year old)
Inverse: If you are not a teenager, then you are not younger than 20. F (think 7 year old)
Contrapositive: If you are not younger than 20, then you are not a teenager. T
Rewrite as both a conditional and converse statement.
A figure is a quadrilateral iff it has four sides.
Conditional: If a figure is a quadrilateral, then it has four sides.
Converse: If a figure has four sides, then it is a quadrilateral.
Create the truth table for p -> q
p q p -> q
T T T
T F F
F T T
F F T
Make a conclusion from the given statements.
If you practice tennis every day, then you will become a better player.
Colin practices tennis every day.
Colin is becoming a better player.
A two-column proof has column headers that say ...
Statement and Reason
What are the next three terms of this sequence?
M, T, W, T, F, ...
These are days of the week
S, S, M
Rewrite as a conditional statement.
All motorcyclists wear helmets.
If you are a motorcyclist, then you wear a helmet.
Rewrite each definition as a biconditional. Is the definition a good definition?
A newspaper has articles you read.
You are reading a newspaper iff you are reading articles.
Not a good definition as you can read articles in other mediums as well.
Create the truth table for p OR q.
p q p OR q
T T T
T F T
F T T
F F F
Make a conclusion from the given true statements.
Angles 1 and 2 are supplementary.
If two angles are supplementary, then the sum of their measures is 180 degrees.
The sum of the measures of angles 1 and 2 is 180 degrees.
What do you call information that is stated within the problem?
Given
What are the next three terms of this sequence?
8, 13, 21, 34, 55, 89, ...
144, 233, 377
Rewrite the statement as a conditional statement.
Two nonparallel lines intersect in one point.
If two nonparallel lines intersect, then it is at one point.
Rewrite each definition as a biconditional. Is the definition a good definition?
An oxymoron is a phrase that contains contradictory terms.
A phrase is an oxymoron iff it contains contradictory terms.
Good definition (unless you consider the people that breathe oxygen and are morons...)
Create the truth table for p AND q.
p q p AND q
T T T
T F F
F T F
F F F
Make a conclusion from the given true statements.
If two angles are congruent, then their measures are equal.
If two angles are vertical, then they are congruent.
If two angles are vertical, then their measures are equal.
Prove the vertical angles theorem.
Vertical angles are congruent.
Proof is in your notes!
For each conjecture, determine a counterexample.
When you multiply a number by 3, then the product is divisible by 6.
False. A counterexample is anything which is a multiple of 3 but not a multiple of 2.
So, 9, 27, 81 are all good counterexamples.
Determine the converse, inverse, and contrapositive of the following. Determine the truth value and provide counterexamples if necessary.
If an angle is obtuse, then its measure is greater than 90 degrees.
Converse: If the measure of an angle is greater than 90 degrees, then it is an obtuse angle. F (it could be a straight angle)
Inverse: If an angle is not obtuse, then its measure is not greater than 90 degrees. F (it could be a straight angle.)
Contrapositive: If the measure of an angle is not greater than 90, then the angle is not obtuse. T
What are two different ways to write the propositional logic symbols for a biconditional?
p <--> q
(p -> q) AND (q -> p)
Create the truth table to show that a conditional statement and its contrapositive are logically equivalent.
p q p -> q
T T T
T F F
F T T
F F T
p q ~p ~q ~q -> ~p
T T F F T
T F F T F
F T T F T
F F T T T
Make a conclusion using the given true statements.
If your father buys new gardening gloves, then will work in his garden.
If he works in the garden, then he will plant tomatoes.
If your father buys new gardening gloves, then he will plant tomatoes.
Assume that Q, R, and S are collinear points.
Given QS = 42, QR = x + 3, and RS = 2x, prove that x = 13.
You can do it!
Email me if you are stuck.
For each conjecture, determine a counterexample.
Two angles in a triangle must add up to be greater than 90 degrees since the interior angle sum of a triangle is 180 degrees.
False. If one angle in a triangle is very large (close to 180 degrees) then the other two angles will sum to be less than 90 degrees.
A good counterexample would be a triangle with angle measures 10, 10, and 160 degrees.
Determine the conditional, converse, inverse, and contrapositive of the following. Determine the truth value and provide counterexamples if necessary.
A square has four sides.
Conditional: If a figure is a square, then it has four sides. T
Converse: If a figure has four sides, then it is a square. F (it could be a rectangle)
Inverse: If a figure is not a square, then it does not have four sides. F (a rectangle is not a square yet has four sides)
Contrapositive: If a figure does not have four sides, then it is not a square. T
Which conditional and its converse form a true biconditional?
A) If x > 0, then |x| > 0
B) If x = 3, then x^2 = 9
C) If x^3 = 5, then x = 125
D) If x = 19, then 2x - 3 = 35
A) F, If x = -1
B) F, if x = -3
C) F, conditional statement is not even true
D) TRUE
Create the truth table for (p -> q) AND (q -> p).
p q p -> q q -> p (p -> q) AND (q -> p)
T T T T T
T F F T F
F T T F F
F F T T T
The following are true:
- If Maria is drinking juice, then it is breakfast time.
- If it is lunchtime, then Kira is drinking milk and nothing else.
- If it is mealtime, then Curtis is drinking water and nothing else.
- If it is breakfast time, then Julio is drinking juice and nothing else.
- Maria is drinking Juice.
Which statements MUST, MAY, or are NOT true?
1. Julio is drinking juice.
2. Curtis is drinking water.
3. Kira is drinking milk.
4. Curtis is drinking juice.
5. Maria is drinking water.
6. Julio is drinking milk.
1. Must be true
2. Must be true
3. May be true
4. Not true
5. Not true
6. Not true
Given angle 1 is congruent to angle 4, prove that angle 2 is congruent to angle 3.
Angles 1 and 4 are congruent (given).
Angles 1 and 2 are congruent because they are vertical angles, which are always congruent.
Angle 4 is congruent to angle 2 by the transitive property.
Angle 4 is congruent to angle 3 because they are vertical angles, which are always congruent.
Angle 2 and angle 3 are congruent by transitive property.