Proofs

Question 2

What does P ⇒ Q mean?

Answer 2

Here P is the premise (hypothesis) and Q is the conclusion based on P. So P ⇒ Q means that “P is a sufficient condition for Q” or “Q is a necessary condition for P”.Exs of expressing it are:1. P implies Q2. If P then Q3. Q follows from P4. Q if P5. Q is neccessary for P6. P is sufficient for QSo If P is true, Q is true by some logical deduction; but if P is false, Q may be still be true by another reason.Ex: If P is the statement “it is saturday” and Q is the statement “it is the weekend” then: P ⇒ Q. But statement Q is true on Sundays too, when P is false.

Question 3

What does P ⇐ Q mean?

Answer 3

The reverse of P⇒ Q:Exs of expressing it are:1. Q implies P2. If Q then P3. P follows from Q4. P if Q5. P is neccessary for Q6. Q is sufficient for P

Question 4

What does P ⇔ Q mean?

Answer 4

P is equivalent to Q ie. the 1st statement is true IF AND ONLY IF the second statement is true.Exs of expressing it are:1. P if and only if Q2. P is equivalent to Q3. P implies Q and Q implies P

Question 5

What does the smbol ¬ stand for?

Answer 5

Negation.Exs:1. If the proposition P is true then its negation ¬P is false.2. If the proposition P is false then its negation ¬P is true.Both propostions P and ¬P cannot be true at the same time.

Question 6

Using Negation symbol ¬ what is the inverse of:P ⇒ Q

Answer 6

¬P ⇒ ¬Q

Question 7

What is the converse of P ⇒ Q?

Answer 7

Q ⇒ P

Question 8

How do you get the contropositive statement?

Answer 8

The inverse of the converse.ex: IF P ⇒ Q then the contropositive is:Q ⇒ P then¬Q ⇒ ¬P

Question 9

Is this statement true: |x + y| ≤ |x| + |y|

Answer 9

YES

Question 10

In proof by contradiction, what would the negation of the statement:”if M^(2) = 14 then M is not a rational number”

Answer 10

“if M^(2) = 14 then M is a rational number”

Question 11

Rewrite the statement using the contropositive:”If 7 is not a factor of n^(2) then 7 is not a factor of n”

Answer 11

“If 7 is a factor of n, then 7 is a factor of n^(2)”

Question 12

Have you learned the direct proof for the formula for the sum of geometeric series?

Answer 12

LEARN IT

Question 13

Have you learned the direct proof for the formula for the sum of Arithmetic series?

Answer 13

LEARN IT

Question 14

Do you know the proof of induction for the Binomial Thereom?

Answer 14

LEARN IT