## 20 Apr Mathematical reasoning and nature of proof

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**1. Nature of Truth**

In mathematics we deal with statements that are “True” or “False”. This is known as the “Law of Excluded Middle”. Despite the fact that multi valued logics are used in computer science, they have no place in mathematical reasoning.

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**2. Nature of Mathematical Proof**

A very common question that comes to our mind is “What is the definition of a good mathematical proof?” And the answer seems to be best given by “It convinces you!” Unfortunately this is not very true. Personal certitude has nothing to do with mathematical proof. The human mind is a very fragile thing, and human beings can be convinced of the most preposterous things. A good proof is one that starts with a set of axioms, and proceeds using correct to the conclusion.

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**3. Rules of Inference**

The common rule of inference that are frequently used are listed below.

1) Given the statement: ~ and the statement , we conclude that

For example:

If I do not wake up, then I cannot go to work.

If cannot go to work then I will not be paid.

Therefore, if I do not wake up, then I will not get paid.

2) Given . We conclude that .

For example:

All cows are Animals, therefore some animals are cows.

An incorrect inference is to conclude that given . After all, not all animals are cows!

3) Given , we can conclude nothing.

For example:

Some cows are Jerseys,

Some Jerseys are human.

Here we interpret the word “Jersey” as “Things that come from Jersey, an island in the English Channel.”

4) Given , we conclude that .

For example:

Some cows are Jerseys, therefore some Jerseys are cows.

5) Given we conclude that .

For example:

Some cows give milk, All things that give milk are female.

Therefore , Some cows are female.

6) Given and . In this case we can conclude nothing.

For example:

All cows are animals. Some animals are birds.

No conclusion is possible.

Now such logical inferences can be formulated in rigorous mathematical format by the proper use of

⦁ A statement such as is said to be . In other words, it is a universal statement that applies to all

⦁ A statement such as is said to be . In other words , there exists at least one to which the statement applies.

⦁ The only permissible form for the universal negative is . The existential negative has several forms like –

Not all A is B

Some A is not B, and many others.

Mathematical statements require somewhat greater precision than general statements.

**4. Negation of a statement**

A proposition is a statement that can be assigned the value or . Negation of a statement is the one that produces a value of when the original statement is and vice versa. In ordinary logic

* An existential negates a universal and a universal negates an existential.

* The negation of is .

* The negative of is .

* The statements and can both be true.

**5. Logical Connectives**

1) If is a proposition, is its negation. is read as .

**Note:** Do not confuse this mathematical connective with the general statement . They are not the same thing.

2) If and are propositions,

* is called the conjunction of and , and is read as .

* is called the disjunction of and , and is read as .

* is called the implication of and and is read as .

**6. Implications**

◦ The most interesting connective is the implication . It can also be written as .

◦ If is false then the entire statement is true. That is .

◦ An implication is proven by assuming that is true and in that case, must also be true.

◦ Given a statement of the form , the statement is called the of .

◦ The Converse of is an independent statement and must be proven independently of .

◦ A statement and its contrapositive are logically equivalent. Either both are true or both are false.

◦ Given a statement of the form , the statement is called the of .

◦ The statement is called the if . The Inverse of is logically equivalent to the Converse of .

◦ The statement of the form is the shorthand for and . In symbols we express this as .To prove , we must prove both and .

**7. Negating Compound Statements**

⦁ X is less than three and X is odd

⦁ X is greater than or equal to 3 or X is even

⦁ The car was either red or green

⦁ The car was not red AND it was not green

⦁ If a person has a Ph.D. then they must be rich

⦁ Prof. Maurer has a PhD and Prof. Maurer is poor.

⦁ Note change in quantifiers.

**8. Rules of inferences**

✓ If is known to be true , is false, and vice versa.

✓ If is true, then is true.

✓ If is true, then both and are true.

✓ If is false and is known to be true, then is false.

✓ If is true, then is true.

✓ If is false, then both and are false.

✓ If is known to be true, and is known to be false then is true.

✓ If is known to be true, and is true then is true.

✓ If is known to be true, and is false then is false.

✓ If is known to be true and is true then is true, and vice versa.

✓ If is known to be true and is false then is false, and vice versa.

✓ If is known to be false and is true then is true, and vice versa.

✓ If is known to be false and is false then $Q$ is true, and vice versa.

9. Logical Fallacies

Most students have a hard understanding this. It is not the calculations that are incorrect, it is the that is wrong. If an inference technique can be used to prove a silly nonsense then it cannot be used to prove anything true. A mathematical proof is actually supposed to demonstrate what is true and apply the rules of inference correctly. So, the next time you write a proof, use proper tools i.e., and do avoid !

Tarun Kumari, Research Scholar,

Dept of Mathematical Sciences,

Tezpur University.

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