RSN Unit-Reasoning and logic
In this unit....
RSN 01
A conditional statement is a statement that can be written as an if-then statement.
Ex.) if p, then q OR If it is a bicycle, then it has 2 wheels.
The hypothesis comes after the "if" and the conclusion comes after the "then"
So if we looked at the statement again...
If it is a bicycle, then it has 2 wheels.
"It is a bicycle" is the hypothesis and "it has 2 wheels" is the conclusion
-Another example
Ex.) if p, then q OR If it is a bicycle, then it has 2 wheels.
The hypothesis comes after the "if" and the conclusion comes after the "then"
So if we looked at the statement again...
If it is a bicycle, then it has 2 wheels.
"It is a bicycle" is the hypothesis and "it has 2 wheels" is the conclusion
-Another example
RSN 02
Given a statement, rewrite it in proper If-Then conditional form
Example: On Tuesday, play practice is at 6.
If it is Tuesday, then play practice is at 6.
Example: On Tuesday, play practice is at 6.
If it is Tuesday, then play practice is at 6.
RSN 03
Logic Chain
Example: A. If cold, turn on heat
B. If sunny day, turn on AC
C. If turn on AC, then it is cold
This could be rewritten as:
-If sunny day, turn on AC
-If turn on AC, then it is cold
-If cold, turn on heat
This would lead to a resulting conditional statement of "If sunny day, then turn on heat."
This chain can also be expressed as if A=B, B=C, and C=D, the A=D
Example: A. If cold, turn on heat
B. If sunny day, turn on AC
C. If turn on AC, then it is cold
This could be rewritten as:
-If sunny day, turn on AC
-If turn on AC, then it is cold
-If cold, turn on heat
This would lead to a resulting conditional statement of "If sunny day, then turn on heat."
This chain can also be expressed as if A=B, B=C, and C=D, the A=D
RSN 04
Given a conditional statement, know how to write the converse, inverse and contrapositive statements.
RSN 05
Use converse, inverse, and contrapositive to determine the validity of conclusions made in association with conditional statements (true/false)
RSN 06
Know how to write a biconditional, given a conditional statement, and state its validity
Helpful video
RSN 07
Determine whether a definition is good or not.
A. If the conditional and converse are true, then all 4 statements (conditional, converse, inverse, and contrapositive) are true.
B. If all 4 are true, then the biconditional is true.
C. If the biconditional is true, then it's a good definition.
So....
If the conditional and converse are true, then it's a good definition.
OR
A definition is good iff the conditional and converse are true.
B. If all 4 are true, then the biconditional is true.
C. If the biconditional is true, then it's a good definition.
So....
If the conditional and converse are true, then it's a good definition.
OR
A definition is good iff the conditional and converse are true.
RSN 08
Determine the "opposite" of a given statement (a.k.a. logical negation)
Negation is saying the OPPOSITE of a statement. For example: -the opposite of day is everything BUT day (it is NOT night) -the opposite of hot is NOT hot Steps: 1.Given a conditional: "I I flip a coin, then it comes up heads." 2.you would keep the hypothesis the same and then using logical negation you would write: "If I flip a coin, then it does NOT come up heads. An example using numbers is: If x<7, then 2x<14 If x<7, then 2x≥14 Here's a video to explain it more. |
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RSN 09
Using Euler Diagrams to demonstrate whether statement is contradiction or not
Here is an example:
Here is an example:
And here is an example I created:
RSN TT-Truth Tables
First, here is a video explaining an intro to Truth Tables
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Definition: A truth table is a breakdown of a logic function by listing all possible values the function can attain.
Intro:
To start off, you should know that true and false are opposites, so if P is true then Not P (~P) is false.
A conjunction (AND=^) requires both values (p&q) to be true. If either is false, then the conjunction is false.
Disjunction (OR=∨) requires that either one is true. If neither is true, then the disjunction is false.
Example:
Intro:
To start off, you should know that true and false are opposites, so if P is true then Not P (~P) is false.
A conjunction (AND=^) requires both values (p&q) to be true. If either is false, then the conjunction is false.
Disjunction (OR=∨) requires that either one is true. If neither is true, then the disjunction is false.
Example:
Conditional Statements in Truth Tables
-A conditional statement is valid if it's not violated. It is only violated if P is true and Q is false.
-The same applies for the converse, except Q and P switch places (violated if Q is true and P is false)
-The inverse is when the opposite of P and the opposite of Q switch places with P and Q-same rules apply as in the conditional statement
-The contrapositive is when the opposite of P and Q are flipped
-A conditional statement is valid if it's not violated. It is only violated if P is true and Q is false.
-The same applies for the converse, except Q and P switch places (violated if Q is true and P is false)
-The inverse is when the opposite of P and the opposite of Q switch places with P and Q-same rules apply as in the conditional statement
-The contrapositive is when the opposite of P and Q are flipped
a) When the Conditional is True, the Converse is sometimes True.
b) When the Conditional is True, the Inverse is sometimes True.
c) When the Conditional is True, the Contrapositive is always True.
b) When the Conditional is True, the Inverse is sometimes True.
c) When the Conditional is True, the Contrapositive is always True.
If you want to test your knowledge of truth tables, use this site
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RSN 10
Indirect Proof-Given a set of conditions and related statements, apply indirect reasoning to come to a conclusion.
If we assume that the conditional (if A then B) is true, then we know that the contrapositive (if not B then not A) is also true. Since the converse (if B then A) and inverse (if not A then not B) are only sometimes true, their truth value is unreliable so they have no conclusion.
-Example: if it rains, then flowers live
a) flowers live ∴ no conclusion (converse)
b) it rains ∴ flowers live (conditional)
c) it didn't rain ∴ no conclusion (inverse)
d) flowers dead ∴ didn't rain (contrapositive)
-Example: if it rains, then flowers live
a) flowers live ∴ no conclusion (converse)
b) it rains ∴ flowers live (conditional)
c) it didn't rain ∴ no conclusion (inverse)
d) flowers dead ∴ didn't rain (contrapositive)
Proof by Contradiction (Indirect Proof)
1) Assume the opposite: if p then not q=p→~q
2) Take all possibilities left (not q=~q) and reason to contradiction (p and not p=p^~p)
3) Since if p then not q (p→~q) can not be true, p then q (p→q) must be true
-Example:
1) Assume the opposite: if p then not q=p→~q
2) Take all possibilities left (not q=~q) and reason to contradiction (p and not p=p^~p)
3) Since if p then not q (p→~q) can not be true, p then q (p→q) must be true
-Example:
- If it's a triangle (Δ), then it has at most 1 right angle (∟)
- Assume: if it's a triangle then there are 2 or 3 right angles (not 1 right angle could only be 2 or 3 since a triangle, by definition, only has 3 angles)
- If it's a triangle, then the angles' sum must add up to 180 degrees.
- If there are 2 angles, then 90 + 90 + 0 = 180
- 0 degrees is not possible, so 2 angles is invalid
- If there are 3 angles, then 90 + 90 + 90 = >180
- having a triangles' angles add up to more than 180 degrees is simply impossible
- Therefore:
- If it's a triangle, then it has at most 1 right angle.
RSN 11
Correctly match Properties of Equality and Congruence with representative examples.
RSN 12
Given an algebraic equation, solve using a two column proof format.
We already know how to solve an equation algebraically. But in geometry, rather than showing your work in the equation, you write out your reasonings. To start off, you have a "given" which is your equation you are trying to solve. You make a table with two columns, one for statements and the other for reasons. The statements column will have all of your equations, but it won't show any of the steps to how you got there. That's what the reasons column is for. In the reasons column, your first steps is always "given". Then you show how you got to the next step. For example if you subtracted 2 from each side, you would state Subtraction Property of Equality. When you have finished your proof, you always end with QED which shows that you have completed the proof.
We already know how to solve an equation algebraically. But in geometry, rather than showing your work in the equation, you write out your reasonings. To start off, you have a "given" which is your equation you are trying to solve. You make a table with two columns, one for statements and the other for reasons. The statements column will have all of your equations, but it won't show any of the steps to how you got there. That's what the reasons column is for. In the reasons column, your first steps is always "given". Then you show how you got to the next step. For example if you subtracted 2 from each side, you would state Subtraction Property of Equality. When you have finished your proof, you always end with QED which shows that you have completed the proof.
And here is an example:
RSN 13
Fill in the blanks to complete a two column proof of Overlapping Angles Theorem, and Overlapping Segments Theorem