How Do We Review Logic?

Knowing all from what is logic to the biconditional, we are ready to perform some examples and review what have we learned so far.


Logic
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Mathematical Sentence

• 17 - 5 = 12 ------------> T

• 17 +3 = 42 ------------> F

• 0.1 < 1 -----------------> T

Conditional
REMEMBER:

IF...THEN is only FALSEwhen
T implies F.
All other cases are TRUE.

Inverse

• "If it is not raining, then it is not cloudy"
• "If twice Talia's age is not 10, then Talia is not 5 years old"

REMEMBER: Under negation, TRUE becomes FALSE - or - FALSE becomes TRUE. Converse Conditional:  "If the space shuttle was launched, then a cloud of smoke was seen." Converse:      "If a cloud of smoke was seen, then the space shuttle was launched." HINT:  Try to associate the logical CONVERSE with Converse™ sneakers -- think of the two parts of the sentence "putting on their sneakers" and "running" to their new positions. Contrapositive HINT:  Remember that the contrapositive (a big long word) is really the combining together of the strategies of two other words:  converse and inverse. An important fact to remember about the contrapositive, is that it always has the SAME truth value as the original conditional statement. **If the original statement is TRUE, the contrapositive is TRUE.   If the original statement is FALSE, the contrapositive is FALSE. They are said to be logically equivalent. Biconditional REMEMBER: IF AND ONLY IF is TRUE   when both facts are T  or both facts are F.. Problem 1. Which of the following statements is the converse of"If the moon is full, then the vampires are prowling."? Choose one:  If the vampires are prowling, then the moon is full.  If the moon is not full, then the vampires are  prowling.  If the vampires are not prowling, then the moon is not full. **Images and Examples provided by regentsprep.org and my notes from Geometry class.

How do we Solve Logic Problems using Conditionals?

In this post you will learn about 2 more conditionals and one special conditional.


Remembering that a conditional is when you use "If" in front of the hypothesis of the statement and "then" in front of the conclusion, we can continue our investigation through conditionals.


A Converse is when you switch the hypothesis and the conclusion.
             Ex: If I ate 2 tacos for dinner, then I'm full. The converse to this conditional is "If I'm full, then i ate 2 tacos for dinner"

A Contrapositive must be the hardest one because that's the one the Regents use a lot on their exams. A contrapositive is when you negate and switch the hypothesis and conclusion. So you use both inverse and converse. Contrapositives is very easy, its just that sometimes it might get difficult, depending on the statements which is given.

Ex: If i study, then I'll pass geometry. The contrapositive to this conditional is "If I do not pass Geometry, then I didn't study"

Remember:The contrapositive is the mixing of the inverse and the converse. 

Problem: What is logically equivalent to the statement "If it is not raining, then I'm happy"

**Images and Examples provided by regentsprep.org and google.com/images and also my notes from Geometry class.

What is a Mathematical Statement and What is Logic?

Logic as we know it, means knowledge/thinking. People use logic for everything. You need logic to know how to tie your shoes, to go to school/work, even to cook! We use logic everyday of our lives and that's what makes us different from animals: we have logic.


A Mathematical Statement is a statement that can be said to be true or false. In order to perform a mathematical statement we need logic, and I'm pretty sure all human beings have logic.
                     Ex: Congruent angles are angles that measure the same. T or F

Open Sentences is a mathematical statement where there is a "variable"
                    Ex of Variables: She is pretty. "She" is the variable.


Negations is pretty simple and pretty self explanatory, the name clearly says what is about. Negations is when you deny something and say "not". It always have the opposite truth value. Using negations, you still have to use logic and a mathematical sentence.
                      Ex: An owl is a fish. --> F
                       - Negation: An owl is not a fish. --> T


Conjunction is a word that helps connect two or more sentences.

• "and": both statements must be true for the statement to be true
• "or": either statement (or both) can be true for the statement to be true

There are 4 types of logic in math. In this post we will learn about 2 only.


A Conditional is the most frequently used statement in the construction of an argument or in the study of math.
                      Ex: Today is raining. Tomorrow will be sunny. The conditional of these statements are "If today is raining, then tomorrow will be sunny". So you basically add "If" to the hypothesis and "then" to the conclusion.


An Inverse is formed by negating the hypothesis and the conclusion.
                   Ex: Today is raining. Tomorrow will be sunny. The inverse to these statements are "If today is not raining, then tomorrow will not be cloudy"


Problem:      "A trapezoid is a four sided polygon"
                                   True             or                 False


**Images and some of the examples provided by google.com/images and my notes from Geometry class.