Contrapositive and Logical Equivalence
By: Christopher Moncrief
Deborah Allen-Carr
Jasmine Franklin


Contrapositives are a lot easier than you may think. They are the exact opposite of any conditional. For example the contra-positive of pvq is ~qv~p. another feature of contra positives is Logical Equivalence (when two logical statements have the exact same truth value). This only applies to the original conditional of the resulting Contrapositive.


1) P: trees are green
Q: pens use ink
Contrapositive of (p^q)-->p=

2) P: Hair is dead skin
Q: Most people have 5 fingers
Contrapositive of (q-->~p)

3) P: Teamwork makes the dream work
Q: Computers have keyboards
Contrapositive of `~p-->~q

4) P: Dell has 4 letters
Q: At Bronx Prep we wear uniform
Contrapositive of (q^~p)-->~q
P: White boards are whiteQ: Black boards are black Contrapositive of (~p-->q)-->(~q^p)


Do these statements have logical equivalence?
6. p-->q :q-->p
7. ~p-->q : q-->p
8. q-->p : ~q-->q
9. ~q-->q : q-->p
10. p-->p :~p-->q



Conclusion of Contrapositives and Logical Equivalence

In conclusion constrapositives are the combination of both an inverse and a converse. A converse is a compound sentence that interchanges the hypothesis with the conclusion. An inverse is a compound sentence that negates both statements of the conditional. Logical equivalence is like a truth table. It explains how the equivalence of statements has the same exact truth values. This concludes how constrapositives are similar to the logical equivalence of two or more statements. The logical equivalence of constrapositives are the truth value of the original of a conditional.

CONJUNCTIONS ^ DISJUNCTIONS۷



Ashley Holland Pasha Ellis India Robinson




INTRODUCTION

THE TOPIC WE ARE GOING TO DISCUSS IS CONJUNCTIONS AND DISJUNCTIONS. A CONJUNCTION IS TWO COMBINED SENTENCES USING THE WORD “AND”. A DISJUNCTION IS TWO COMBINED SENTENCES USING THE WORD “OR.”. THERE ARE SYMBOLS FOR BOTH CONJUNCTIONS AND DISJUNCTIONS. THE SYMBOL USED FOR A CONJUNCTION IS ٨WHICH STANDS FOR “AND”. THE SYMBOL USED FOR DISJUNCTIONS IS V WHICH STANDS FOR “OR”. AN EXAMPLE OF A CONJUNCTION IS, LISA IS 5 YEARS OLD AND JAMES IS 2 YEARS OLD. AN EXAMPLE OF A DISJUNCTION IS LISA IS 5 YEARS OLD OR JAMES IS 2 YEARS OLD.




State the truth values for the following if p is true and q is false and if r is false & s is true.

P: the sky is blue r: tomorrow is Monday
Q: the grass is green s: Monday is the 5

1. The sky is blue AND the grass is green. true AND false =FALSE
2. The sky is blue OR the grass is green. True OR false=TRUE
3. Tomorrow is Monday OR Monday is the 5. false OR true= FALSE
4. Tomorrow is Monday AND Monday is the 5. false AND true=FALSE
5. The grass is green OR Monday is the 5. false OR true=FALSE



Create a truth tables for the following:

1. p^q ~p^q
































2. p q















3. q^ p p^ q

























4. p^q















5. q^ p















6. ~p^ ~q
































7. ~q^p ~p^q






























8. ~q^p




















9. ~q p




















10. ~p ~q



































CONCLUSION

AS YOU CAN SEE WE USE CONJUCTIONS AND DISJUNTIONS EVERYDAY. THEY ARE APART OF OUR LIVES. YOU CAN USE ENGLISH AND MAKE IT INTO A MATH STATEMENT. CONJUNCTIONS AND DISJUNCTIONS ARE VERY USEFUL AND EASY BUT DID YOU SOLVE THESE PROBLEMS CORRECTLY? THE EASIEST THINGS MAY BE THE HARDEST TO SOLVE!!!!!!!!!!

Monaisia Livingston, Nikiray Colon

DeMorgan’s Law & Law of Detachment

De Morgan’s Law also known as DeMorgan’s Theorem is a law that deals with double negation. The relationship that deals with the double negation is known as DeMorgan’s duality. When DeMorgan’s Law is applied the conjunction or disjunction becomes the opposite. The Law of Detachment states that if the conditional is true than the hypothesis is true. If the hypothesis is true then the conclusion is true. Both laws can be represented symbolically.

1. ~ (~P^~Q) = P or Q
2. ~ (~P^Q) =P or ~Q
3. ~ (~P or ~Q) =P^Q
4. P→Q
P___

5. Q→P
Q______




1. ~(~p ^~(~Q))=
2. ~(~Q v ~(~P))=
3. P-Q
P__
4. Q ^~p=
5. ~(~Q v ~P)=
6. ~(Q v~(~P))=
7. p ^~Q=
8. ~Q v ~P
9. Q-P
Q______

10.Q-s
Q_

Conclusion: This is demorgan’s law and Law of detachment. Now you know how it works so show your magic and solve these problems. You shouldn’t have trouble now that you’ve seen the examples and read the intro. You always want someone to tell you the truth so think of that as law of detachment and just know that this law is always true

!! SHAMiRA.....JAELEEN......STEPHANiE !!

TRUTH TABLES

Truth tables are very important in logic. They are used to keep things organized. Also they are used to figure out equations correctly. Without truth tables people would get confused and wouldn’t know how to keep track of their work. Truth tables help figure out if the problem is true or false. Without truth tables people wouldn’t be able to do logic properly.


Directions: Create truth tables for these 10 equations

q: true
p: false

1.~q -> p
2.p -> q
3.~q -> ~p
4.p ^ q
5.~q v p
6. q ^ ~p
7. (p v q) -> (~q ^ p)
8.p ^~q
9.((~q ^ p) -> (q ^ ~p)
10. p v ~q

To conclude truth tables are used to show how something can be proved. In math it shows how statements can be switched around and still be true. When true statements are negated and/or is switched around the conclusion of the statement may change. The examples that were shown may help you better understand what truth tables are for. These tables may have different types of equations that come out to different conclusions.

BICONDITIONALITY
BY: Steven Fernandez
David Guity
And. Billy Wilson dundundunnnnn…

Biconditionals are when 2 compound logic statements are put together using “and” or “if and only if”. We use biconditionals so that lazy mathematicians can relate information to each other with out having to write a lot. In a Biconditional there will be many symbols such as ~(negative) ^(and) v(or) and -->(conditional). An example of a biconditional is:

(P^Q)<-->(PVQ) If P was true and Q was false than:
(T^F)<-->(TVF) You first solve what is inside the parenthesis:
F<-->T than u find the answer of the bicontitional
False! Tip! If at the end there are 2 different values (T+F or F+T) the answer is false, if there are 2 of the same values (T+T or F+F) than the answer is true.

Examples:
If P is true and Q is false then solve these equations:
1) ~P<-->Q= F<-->F=T
2) P<-->Q=T<-->F=F
3) (Q^P) <-->(P^Q) = (F^T) <-->(T^F) =F^F=T
4) (P^Q) <-->Q= (T^F) <-->F=F<-->F=T
5) P <-->(QvP) =T<-->(FvT)=T<-->T=T


Class Problems:
If P is false and q is true then solve these equations:

1. P<-->q
2. ~p<-->~q
3. (~q<-->~q) -->p
4. (~p^q) <-->q
5. (P<-->q) ^p
6. (~p<-->~q) ^q
7. (P<-->q) -->q
8. ~p<-->q
9. P<-->~q
10. (~p<-->q) -->p

Conclusion:
Biconditions are basically a double conditional. . In a Biconditional there are many symbols such as ~(negative) ^(and) v(or) and -->(conditional). Biconditionals are when 2 compound logic statements are put together using “and” or “if and only if”.

Remember: We use biconditionals so that lazy mathematicians can relate information to each other with out having to write a lot.

Isamar Story

Shamira was going on a baby roller coaster with her niece. At first the roller coaster started slow. Then it went a little bit faster. Then at the top of the hill of the roller coaster it started to slow down. Then it reached a point where it almost completely stopped. Then it started going faster then the first time. Then it dropped again a little bit faster. Then it slowed down again but suddenly when really fast up the hill. Then it started to slow down and the roller coaster was over.

Isamar Story

Shamira was going on a baby roller coaster with her niece. At first the roller coaster started slow. Then it went a little bit faster. Then at the top of the hill of the roller coaster it started to slow down. Then it reached a point where it almost completely stopped. Then it started going faster then the first time. Then it dropped again a little bit faster. Then it slowed down again but suddenly when really fast up the hill. Then it started to slow down and the roller coaster was over.

Shamira's Story

One morning Isamar was going to school. She was going on her way to school but then she thought she forgot her cell phone. When she was near her house she realized she had it the whole time. She went back to school and when she got there she got sent back home because she was out of uniform. Isamar was on her way again to school but when she was almost there her mother called her and told her not to go to school because she wasn’t feeling well so she went home.

NIKIRAY COLON
DAVID GUITY





DISTANCE









TIME


Jackie Chan had a speed talisman (a type of rock that has special supernatural powers) and started running from the shadowcon (an evil organization’s henchmen), while chewing his gum. He suddenly tripped and fell. He retraces his steps trying to look for the talisman, when suddenly the shadowcon start to attack Jackie. He does some of his kicking and jumping and flipping. One of the shadowcon has the talisman and starts to beat up Jackie Chan. It’s going so fast Jackie Chan can not defend. Jackie Chan realizes that he must trap him somehow so he drops the gum he was chewing on the floor. He snatches the talisman and starts to run like the wind. the talisman starts to take it’s toll on Jackie so he starts to slow down.

nikiray colon david guity





speed TIME
One day Jack was steadily walking up the hill. He saw Jill and started to get excited. He started to run a bit faster. Once he met Jill, he suddenly tripped over a rock and started to roll down the hill. He was all bloody and sad when he made it to the bottom. He was crying and sobbing. He felt like he was going to die. He wanted to go and meet Jill since Their Love was strong so he started to climb the hill again. They walked down the hill happily ever after.

What was the hurry??
Ashley & Monaisia

Monaisia lived at the bottom of the hill, down the block and around the corner. One day she was running home and she hit the top of the hill and began to run. While she ran down the hill she began to pick up speed, fell and broke her wrist. She made a sudden stop when she got to the bottom to make sure her wrist wasn’t falling off or nothing. She had to walk diagonally across the street and around the corner to Maple Hill Rd. Then when she got home she realizes that the in jury was bad. When she went to play ball at The UTK they told her she couldn’t because of her wrist!! At the end of the night Monasia said,” I Shoulda Coulda Woulda but I didn’t walk home.”

david guity
nikiray colon



distance



TIME Jackie Chan had a speed talisman (a type of rock that has special supernatural powers) and started running from the shadowcon (an evil organization’s henchmen), while chewing his gum. He suddenly tripped and fell. He retraces his steps trying to look for the talisman, when suddenly the shadowcon start to attack Jackie. He does some of his kicking and jumping and flipping. One of the shadowcon has the talisman and starts to beat up Jackie Chan. It’s going so fast Jackie Chan can not defend. Jackie Chan realizes that he must trap him somehow so he drops the gum he was chewing on the floor. He snatches the talisman and starts to run like the wind. the talisman starts to take it’s toll on Jackie so he starts to slow down.

BillyWilsonAndStevenFernandez.com

my first diving lesson


done by chris and schenell

POSTING REAL LIFE GRAPHS on the BLOG

For visionaries who have finished the mini-project on real life graphs, you should now post your graph and story on the blog.

In order to post a picture on the blog, you must do the following:

1. Copy and past your graph into Paint (unless it is already there).
2. Go to "Save" in PAINT
3. Go to "Save as Type" and select JPEG (this must be used, otherwise the file will be too big and NOT post on the blog).
4. Name your picture document, and save it on the student server.
5. Log onto www.blogger.com
6. Type in your username and password.
7. Select 'Create New Post'.
8. Type your name and your partner's name, and hit Enter.
9. Click on the picture icon, showing a blue sky.
10. For "Choose a Layout", select "None".
11. For "Image Size", select "Large".
12. Select "Browse" from the menu "Add an image from your computer".
13. Find the Paint document you just saved and select it.
14. Hit the blue button "Upload Image".
15. Underneath the image, copy the story from Word.
16. Edit and make any changes to your posting before publishing it.
17. Save the Draft if you are not finished, or "Publish Post" if you are.
18. Finally, comment on each other's graphs with constructive comments.

Happy Blogging!

REAL-LIFE GRAPHS MINI-PROJECT

Hello visionaries!!! Check out the cool new look for our blog (in honor of V-day)!

Today in the lab you will be drawing a real-life graph in Microsoft Word or Paint. It can be the graph you sketched in class yesterday, or a brand new one you create. Just remember: time cannot go backwards, so make sure your graph is logical.

To create your graph in Word, right click anywhere in the blank space below the blue bar, and make sure the Drawing menu has a check next to it. Then, all the tools you will need to draw your graph will appear at the bottom of Word. EACH partner should draw their OWN graph.

For your graph, make sure you provide the following:
*y-axis title
*x-axis title
NO GENERAL TITLE...your partner will provide that.

Once your graph is drawn and you have saved your work, switch documents with a partner. It is up to the partner to create a story that describes their partner's graph accurately. The story should be typed BELOW the graph. Higher grades will be given to stories that are very creative and that match the given graph.

Once you are finished, print out your graph and story with your name and your partner's name at the top, and submit it to Mrs. Collins. Then, you need to work on posting your work on the blog. You will have two days in lab to complete the project.

GOOD LUCK ON THE MID-TERMS

HAPPY HOLIDAYS!!! HAPPY VACATION!!!

Have a great break (remember to complete the packet), and I'll see you in 2007!

CU,
Mrs. Collins