1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Korvikt [17]
3 years ago
8

Prove that P (P) = (QA ~ Q)] is a tautology.

Mathematics
1 answer:
alekssr [168]3 years ago
8 0

Answer:

The statement P \leftrightarrow [(\lnot P) \rightarrow (Q \land \lnot Q)] is a tautology.

Step-by-step explanation:

A tautology is a formula which is "always true" that is, it is true for every assignment of truth values to its simple components.

To show that this statement is a tautology we are going to use a table of logical equivalences:

P \leftrightarrow [(\lnot P) \rightarrow (Q \land \lnot Q)] \equiv

\equiv (P \land [(\lnot P)\rightarrow (Q \land \lnot Q)]) \lor(\lnot P \land \lnot [(\lnot P)\rightarrow (Q \land \lnot Q)]) by the logical equivalences involving bi-conditional statements

\equiv (P \land [\lnot(\lnot P)\lor (Q \land \lnot Q)]) \lor(\lnot P \land \lnot [\lnot(\lnot P)\lor (Q \land \lnot Q)]) by the logical equivalences involving conditional statements

\equiv (P \land [P\lor (Q \land \lnot Q)]) \lor(\lnot P \land \lnot [ P\lor (Q \land \lnot Q)]) by the Double negation law

\equiv (P \land [P\lor (Q \land \lnot Q)]) \lor(\lnot P \land \lnot P\land \lnot(Q \land \lnot Q)) by De Morgan's law

\equiv (P \land [P\lor F]) \lor(\lnot P \land \lnot P\land \lnot(Q \land \lnot Q)) by the Negation law

\equiv (P \land [P\lor F]) \lor(\lnot P \land \lnot P\land \lnot Q \lor \lnot(\lnot Q)) by De Morgan's law

\equiv (P \land [P\lor F]) \lor(\lnot P \land \lnot P\land \lnot Q \lor  Q) by the Double negation law

\equiv (P \land P) \lor(\lnot P \land \lnot P\land \lnot Q \lor  Q) by the Identity law

\equiv (P) \lor(\lnot P \land \lnot P\land \lnot Q \lor  Q) by the Idempotent law

\equiv (P) \lor(\lnot P \land \lnot P\land  (Q\lor \lnot Q)) by the Commutative law

\equiv (P) \lor(\lnot P \land \lnot P\land T) by the Negation law

\equiv (P) \lor(\lnot (P \lor P)\land T) by De Morgan's law

\equiv (P) \lor(\lnot (P)\land T) by the Idempotent law

\equiv (P \lor\lnot P) \land(P \lor T) by the Distributive law

\equiv (T) \land(P \lor T) by the Negation law

\equiv (T) \land(T) by the Domination law

\equiv T

You might be interested in
David invests $7000 in two different accounts. The first account paid 4 %, the second account paid 8 % in
Irina-Kira [14]

What we know:

I  = Invest (so we dont mess up on interest.)

So, I = $7,000

First Account is 4%

Second Account is 2%

End of the year David Earns $476 in interest

With percent's you can always turn them into a decimal.

--------------------------------------------------------------------------------------------------------------

The Question: How much was in each account?

--------------------------------------------------------------------------------------------------------------

Step by Step Answer:

I = amount invested at 4%

4% = 0.40

8% = 0.80

0.40 - $7,000 = 6999.6

0.80 - 6999.6 = 6998.8

6998.8 - 476

Amount in both accounts in total is 6522.80.

--------------------------------------------------------------------------------------------------------------

HOPE THIS HELPS! =3

HAVE A FANTASTIC DAY!

6 0
3 years ago
Someone please help me:// with A,B,andC...if you don't know don't answer please
iVinArrow [24]
For number one the second card must be positive or he would have to put it back if its negative for number 2 the card must be 3 4 5 because negative 2 plus 3 is 1 which is positive and negative 2 plus 4 and 5 would allow her to keep her cards. for the third one danny could have pulled a negaitive 1 2 3 4 and 5. oh dude i forgot to put decimals but still the same thing
3 0
2 years ago
PLEASE HELP I REALLY NEED HELP
Rainbow [258]

Answer:I believe that your asnwer is c.

Step-by-step explanation:

This is because -4 + 4 = 0 10 - 10 = 0 7-7 = 0

5 0
3 years ago
Keith’s high school played 4 football games this year, 2 of the games were played at night . He attended 3 games. How many footb
Nikitich [7]

Answer:

1

Step-by-step explanation:

because there were 4 games playes in total and he attended 3 of the 4 games so 4-3=1

4 0
3 years ago
- 7th Grade Work -<br><br> Please write your answer like this:<br><br> 1.<br> 2.<br> 3.
Naily [24]

Answer:

1. -4x < 8

2. 4x < 8

3. When something is done to one side of the equation, the same thing must be done to the other side. Dividing each side by -4 in part A would cause the sign to flip because the sign is always flipped when dividing by a negative. Because 4 in 4x (part B) is not negative, the sign is not flipped.

Step-by-step explanation:

Hope this helped!!

8 0
3 years ago
Read 2 more answers
Other questions:
  • Which ordered pair is a solution of the equation?
    7·1 answer
  • You deposit $2000 in an account earning 4% interest compounded continuously. How much money will you
    11·1 answer
  • Find four solutions to the equation (x^2-9)(x^4-16)=0
    15·1 answer
  • Which expression is equivalent to 1.18 x 0.6
    7·1 answer
  • Solve 2x - 1 &lt; 3<br> a) x&gt;1<br> b) x&gt;2<br> c) x&lt;2<br> d) x&lt;1<br> e) x&lt;-2
    8·1 answer
  • What is the range of the funtion?<br>​
    8·1 answer
  • on Monday the number of customers in the store rounded to the nearest hundred was 400 what is the greatest number of customers t
    6·2 answers
  • The perimeter of a rectangle is 104 meters. If the length of the
    13·1 answer
  • Which of the Following is an example of a claim?
    10·1 answer
  • Please help!!!!!!
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!