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.