Download Counter Example

Gaby Gottlib
Journal 2
A conditional ifthe statement is a statement that has two
parts: PQ
P: Hypothesis
Q: Conclusion
If I do my homework then I will go to the
movies.
P

Q
Examples:
1. If I clean my room then I will go to the party.
2. If I feed my pet then I will buy another one.
3. If I study for the test the I will get 100%.
Same as conditional statement but P and Q
are NOT
PQ
If I don’t do my homework
then I don’t go to the movies.
When you flip the P and the Q of your
conditional statement.
QP
If I go to the movies then I did my
homework.
Negative the converse
QP
If I don’t go to the movies then I
didn’t my homework.
A counter example is one example that disproves a hypothesis.
Examples:
1. All prime numbers are odd
Counter example: 2
2. All animals living in the ocean are fish
Counter example: whales
3. All the people in the class are girls.
Counter example
1. Collect data
2. Look at the facts
3. Making conclusions
Deductive reasoning is when you look at facts that
have happened before to make a conclusion
Examples:
1. Gravity makes things fall. That is why that apple fell in my head.
2. All oranges are fruits. All fruits come from trees. Therefore oranges grow on trees.
3. Earth is a planet. All planets rotate around the sun.
Therefore, Earth rotates around the sun.
Symbolic notation: Is writing expressions with
symbols instead of having to write
them down.
1. Law of Detachment: If PQ is a true statement then if P is true, then Q must be
true also.
o If it’s Friday night then I will go on a date with my wife.
Today is Friday you will be in a date with your wife.
o If I clean my room then I will go to the party.
I cleaned my roomI am going to the party
2. Law of Syllogism: If PQ and QR are both true statements, then if P is true
the R is also true.
o If it is Friday nightI will go on a date with my wife.
If I go in a date with my wifeI will go to the movies.
If it is Friday night then I will go to the movies.
o If I clean my roomI will go to the party.
If I go to the partyI will eat candy.
If I clean my room then I will eat candy.
A definition is when you say what something is. All definitions are bi-conditional
Examples:
1. Marker: An object that has ink inside and is used to draw on a whiteboard
2. Pig: A farm animal that is usually dirty
3. Computer: Electronic device that helps us do things in an easier way
A bi-conditional statement is when both a conditional and a converse statement are true.
IF AND ONLY IF=iff
They are used when both the conditional and the converse statements are true.
They are important because it is an easier way to write them both.
1. A shape is a triangle IFF it has three sides.
Conditional: If a shape is a triangle then it has three sides.
Converse: If it has three sides then it is a triangle.
2. Today is Saturday or Sunday IFF it is the weekend.
Conditional: If today is Saturday or Sunday then it is weekend.
Converse: If it is the weekend then it is Saturday or Sunday.
3. It is a school day IFF it is Monday.
Conditional: If it is a school day then it is Monday.
Converse: If it is Monday then it is a school day.
Proof: A logical step by step argument that validates your conclusion.
How to do it:
1. They give you an equation
2. Solve it (show all work and explain why can you do each step.)
o 3x-8=19
1st: add 8 to both sides
2nd: divide by 3 in both sides
3rd: say the answer x=9
Addition Property
Division Property
Simplify
o 4x-2=6x+8
1st: add 2 to both sides
2nd: subtract 6 from both sides
3rd: Divide by -2 on both sides
4th: say the answer x=-5
Addition Property
Subtraction Property
Division Property
Simplify
o 6x-3=15
1st: add 3 to both sides
2nd: divide by 6 in both sides
3rd: say the answer x=3
Addition Property
Division Property
Simplify
Transitive: a=b and b=c  a=c
Reflective: a=a
Symmetric: a=b  b=a
Addition: a=b  a+c= b+c
Subtraction: a=b  a-c+ b-c
Division: a=b  a/c= b/c
Multiplication: a+b  ac= bc
You have to do two columns and label one statement and the other one reason.
Statement
Reason
 Points P,Q,R and S are collinear
PS=PQ+QS
PS-QS=PQ
PQ=PS=QS
Given
Segment addition postulate
Subtraction Property of Equality
Symmetric Property of equality
 Points A,B,C and D are collinear
AD=AB+BD
AD-BD=AB
AB=AD=BD
Given
Segment addition Postulate
Subtraction Property of Equality
Symmetric Property of equality
 3x-9=0
3x=9
X=3
Given
Addition Postulate
Division Postulate
If two angles are a linear pair then they are supplementary.
Examples:
1.
LPP because they are supplementary
1
2
1.
LPP because they add up to 180°
1
2
1.
LPP because they add up to 180° and are
supplementary.
1
2
If two angles are supplementary to the same angle then they are
congruent.
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(0-10 pts) Describe the congruent complements and supplements theorems. Give at least 3
examples of each.___
Vertical angles are always congruent.
1.
2.
1
90°
90°
90°
3.
3
4
2
90°
>1 is congruent
to >3
Transitive: a=b and b=c  a=c
Reflective: a=a
Symmetric: a=b  b=a
Addition: a=b  a+c= b+c
Subtraction: a=b  a-c+ b-c
Division: a=b  a/c= b/c
Multiplication: a+b  ac= bc
If two angles are supplementary to the same angle then they are congruent.
Examples:
1. M<A + M<C = 180° and M<C + M<B = 180°. So, < A and <B are congruent.
1. M<B + M<D = 180° and M<D + M<C = 180°. So, < B and <C are congruent.
3. M<G + M<I = 180° and M<I + M<H = 180°. So, < G and <H are congruent.
If points A, B, C, and D are all collinear, then segment AB is congruent to segment CD then
segment AC is congruent to segment BD.
EXAMPLES:
From Michelle’s house to Gaby is the same as from Jakes house to Marias house. Then from
Michelle's house to Jakes house is the same as from Gaby's house to Marias house.
From New York to L.A is the same as from Miami to Denver. Them from New York to Miami
is the same and from L.A to Denver.
From Guatemala to Mexico is the same as from El Salvador to Nicaragua. Them from
Guatemala to El Salvador is the same as from Mexico to Nicaragua.