Mathematical Logic – Mathematica Education

Before we enter the mathematical logic, we must first know the definition of logic which will later play a very role in understanding the logic of mathematics itself. Logic comes from the ancient Greek word λόγος (logos) which means the result of the consideration of the mind expressed through words and expressed in language. Logic has several benefits, namely:

• Helping everyone who studies logic to think rationally, critically, straight, fixed, orderly, methodically and coherent.
• Improve abstract, careful, and objective thinking ability.
• Increase intelligence and increase the ability to think sharply and independently.
• Forcing and encouraging people to think for themselves by using systematic principles
• Increase the love of truth and avoid mistakes, confusion, and mistakes.
• Able to analyze an event.
• Avoid occult, gugon-tuhon (Javanese)
• If you are able to think rationally, critically, straight, methodically and analytically as stated in the first item, it will improve one’s self -image.

After we know about our logic it will be easier to learn mathematical logic. The following are matters relating to mathematical logic.

1. Statement

What is meant by a statement is a sentence that has a right or wrong value but not at the same time both (right and wrong). And a sentence is not a statement if we cannot determine the sentence right or wrong or contain relative understanding. There are two types of mathematical statements, namely closed statements and open statements. A closed statement is a statement whose truth value is certain while an open statement is a statement whose truth value is uncertain. For more details, consider the example below.

Arispersitemens
Example:

6 × 5 = 30 (correct closed statement)

6+5 = 10 (False Closed Statement)

White sugar tastes sweet (open statement)

The distance between Jakarta Bandung is close (not a statement, because close is relative)

2. Denial statement (negation)

The denial is a statement that denies given. Decision of statements can be formed by adding ‘not true that …’ in front of the statement that is denoted ~.

Example:

statement B: two -wheeled motorcycle

Negate statement B: incorrect two -wheeled motorcycle

3. Compound statement

3.1. Conjunction

a statement P and Q can be combined with conjunctions ‘and’ so as to form a compound statement ‘P and Q’ which is called the conjunction is symbolized by

The table besides shows several statements that are combined into conjunction compound statements.

If you find a statement, we just pair it with a table beside so that we can find how compound conjunctions are.

3.2. Disjunction

a statement p and q can be combined with the word connecting ‘or’ so as to form a compound statement ‘P or Q’ which is called the disjunction symbolized by

The table besides shows several statements that are combined into compound sentences of disjunction.

So if we find a statement and we will make a compound sentence we just look at the table, find which one is suitable then we will find how the form of compound sentences.

3.3. Implications

a statement P and Q can be combined with the word CONTACT ‘If then’ so that it forms a compound statement ‘Jikap then Q’ which is called the implications and is symbolized by

The table besides shows several statements that are combined into compound sentences implications.

So if we find a statement and we will make a compound sentence our implications just look at the table beside, find what is suitable then we will find how the compound sentences are implications.

3.4. Bouquet

a statement p and q can be combined with the word connecting ‘if and only if’ so that it forms a compound statement ‘p if and only if q’ which is called biimplication and symbolized by

The table besides shows several statements that are combined into biimplication compound sentences.

So if we find a statement and we will make a biimplication compound sentence we just need to see the table beside, find which one is suitable then we will find how the form of compound sentences of biimplikasinya. Then we will more easily solve the problems that we will face later.

4. Equivalent to compound statements

The equivalence of these multiple statements is very important. We must know the form of negation from conjunctions, negation from disjunction and so on in completing various forms of statement that will later appear. So we must memorize the form of euivalence of multiple statements besides. Then we will more easily solve various types of questions that we will find later. It would be nice for us to memorize the equivalence of the statements beside.

No need to be confused and burdened, the key to mathematics is memorized the formula and can use it. If we often practice questions, we will automatically memorize, and of course we will easily use the formula if applied in the problem.

5. Conversion, inverse and contraposition

From an implication, a statement called conversion, inverse and contraposition can be reduced

6. Kuantor statement

The statement of the head of the office is a statement that contains a quantity measure. There are 2 types, namely:

6.1 Universal Quantor

In a universal quantor statement there is an expression that states all, each. Universal quantor is symbolized by ∀ (read for all or for each).

Example: ∀ x ∈ R, x> 0 read for each x member of real numbers then applies x> 0.

6.2 Existential Quantor

In the existential quantor statement there is an expression that states that there is, some, in part, exist. Existential quantors are symbolized by ∃ (read there, some, some, part)

Example: ∀ x ∈ R, x+5> 1 read there is x member of the real number where x+5> 1.

7. Denial of the statement of the office

The denial of a universal headline statement is a statement of existential headwantor, and vice versa the denial of the existential -re -office statement is a statement of universal headwant.

Example:

P: Some high school students are diligent in learning

~ P: All high school students are not diligent in learning

8. Drawing conclusions

Conclusions are made from several statements that are known to the truth value called the premise. Then by using the existing principles obtained a new statement called conclusions/conclusions derived from the existing premise. Such conclusions are often called arguments. An argument is said to be valid if the premises are correct then the conclusions are also true. There are 3 methods in drawing conclusions, namely:

8.1 Placing mode

Press 1: P → Q

Print 2: P (placing mode)

__________________

Conclusion: Q.

The meaning of the ponens mode is “if known P → Q. And Pthen conclusions can be drawn Q.“. For example:

Premise 1: If you come then your sister will be happy

Premise 2: Father Comes

__________________

CONCLUSION: Sister is happy

8.2 Tollens mode

Press 1: P → Q

Print 2: ~ q (retrieval mode)

__________________

Conclusion: ~ p

Tollens mode means “If you know P → Q. and ~Q.then conclusions can be drawn ~P“. For example:

Premise 1: If it rains, then the sister uses an umbrella

Premise 2: Sister does not use an umbrella

___________________

CONCLUSION: DAY NOT RAIN

8.3 Syllogism

Premise 1: P → Q

Premise 2: Q → R (Syllogism)

_________________

Conclusion: P → R

Syllogism means “if you know P → Q. And Q → R.then conclusions can be drawn P → R.“. For example:

Premise 1: If the price of fuel rises, the price of trees rises.

Premise 2: If the price of trees rises then everyone is not happy.

__________________________________________________

Conclusion: If the price of fuel rises, then everyone is not happy.

Additional Note:

Law de Morgan:

¬ (p λ q) ≡ (¬PV ¬Q)
¬ (PV Q) ≡ (¬p λ ¬Q)

Equivalence Implications:

(P → Q) ≡ (¬PVQ)