De Morgan Law

Previously, we get that p  ~(~p). Negation of the other rules we already know is p ^ ~ q ~ (p -> q), which is the negation of a conditional. Whether we can find a formula that is identical, ie, the negation, in another compound statement, ie, disjunction and conjunctions? The answer is yes, the logic put forward by the British mathematician Augustus de Morgan.

De Morgan's rules, so the formulas of negation and disjunction of conjunctions is called, is as follows.

• The negation of a conjunction is given by ~(p ^ q)  ~p v ~q
• The negation of a disjunction is given by ~(p v q)  ~p ^ ~q

Equivalent expression

When you buy a car, the car can be new or old. Sales will say "of course not true this car is not a new car." This compound sentence has a one-sentence statement (This car is new) and two negation.

"Absolutely not true" "This car is not a new car"
~ ~ p

Does this mean a new car? To find out the truth, we can construct the truth table for logic expression ~ (~ p) and compare it with the p. Since there is only one single statement, then we only need 2 lines in the table. We will have a column p, ~ p and ~ (~ p). ~ p gives the value opposite p. Medium ~ (~ p) gives the value in contrast to ~ p. So the table would be as follows.

p	~p	~(~p)
B	S	B
S	B	S

Note that the third column of ~ (~ p) has a truth value is identical to p. If like this, then ~ (~ p) is said is equivalent to p. That is, meaning they are the same statement. An equivalent expression tends to be a symbolic expression that has an identical truth value to each other.

Expression p q read p is equivalent to q or p and q equivalent. From the above table we find that ~ (~ p) is equivalent to p, can we write p ~(~p).

EXAMPLE
Is the statement "If my landlord, who pays my property tax" and the statement "I am the owner of the house and I do not pay property tax" equivalent?

COMPLETION
We begin by giving a symbolic representation of the statements above.
P: My landlord
q: I pay property tax
p -> q: if I am the owner of the house, I pay property tax.
p ^ ~ p: I am the owner of the house and I do not pay property tax.
Truth table for this case will contain four lines. The table we will use as below.

p	q	~q	p^~q	p -> q
B	B
B	S
S	B
S	S

Now we just give the truth value corresponding to each column. Column ~ q, will we fill with the opposite truth value from column q. The next column of the contents of B in the second row and the S on the other line, since conjunction requires the truth value of B for each factor. Meanwhile in the last column, because the conditional is only false if p is true and q is false, then fill the second row with S and B on the other line.

p	q	~q	p^~q	p -> q
B	B	S	S	B
B	S	B	B	S
S	B	S	S	B
S	S	B	S	B

Because the truth values ​​in the column p ^ ~ q is not equal to p -> q, the two statements are not equivalent. Note that p ^ ~ q and p -> q has the truth values ​​of the exact opposite. When this happens in two statements, one compound statement is the negation of another compound statement. Consequently, p ^ ~ q is the negation of p -> q. This relationship can be expressed by p ^ ~ q ~ (p -> q). The negation of a conditional premise is equivalent to conjunction and negation conclusion.

The statement that looks different in reality may have the same purpose. When we have two statements are equivalent, we can replace each other without changing the meaning. Emotional factors that influence the selection of our statement in practice. Not on its meaning.

Truth table of Conditional

Conditional is a compound statement in the form 'if p then q' is denoted p -> q. What is truth conditional? Notice the phrase 'If you give Rp. 20.000, - to me, I bought concert tickets going for you. "Symbolic representation of the sentence is as follows:

p: You gave me Rp. 20.000, -
q: I bought concert tickets for you.
p -> q: If you gave me Rp 20,000, -, I'll buy you a concert ticket.

Conditional can be viewed as a promise. Assuming you really gave me 20 thousand, then I will have two options, whether buying or not. When I bought (q = B) then the statement is true. But if I do not buy (q = S) then the sentence is false. This situation can be described as follows.

p	q	p->q
B	B	B
B	S	S
S	B	?
S	S	?

Then what if you do not give me 20 thousand? p = S? Of course if I would give you a ticket or not, p -> q is not wrong. Because the promise is only fulfilled if p is true. Since p -> q is false, then if p either directly make the truth value of p -> q is true. Table complete conditional is as follows.

p	q	p->q
B	B	B
B	S	S
S	B	B
S	S	B

Truth table of Disjunction

Disjunction of two statements combined with logical conjunctions 'or'. Compound statement "Burhan is a doctor or a cadre of the Golkar Party" is a disjunction (or inclusive) with the symbolic presentation:

p: Burhan a doctor
q: Burhan Golkar Party cadres
pvq: Burhan a doctor or a cadre of the Golkar Party.

Despite the fact that a doctor is not Burhan Golkar Party cadres, disjunction pvq above remain true. Disjunction is true if there are at least one constituent that statement is true. Disjunction of one if the truth value of each constituent wrong.

p	q	pvq
B	B	B
B	S	B
S	B	B
S	S	S

Truth table of Conjunction

Conjunctions combine two statements with logical conjunctions "and." Compound sentence "Burhan a doctor and the Golkar Party cadres" is a symbolic representation sebagia conjunction with the following.

p: Burhan a doctor
q: Burhan democrat party cadres
p ^ q: Burhan a doctor and democrat party cadres.

The value of truth of a statement depends on the truth value of compound statements that constitute them. How many rows are needed in conjunction truth table for p ^ q? Since p has two possibilities, as well as q, the combination of the possibilities that there are 4 (2.2).

p	q
B	B
B	S
S	B
S	B

A conjunction p ^ q is true if each of the statements that constitute true, a combination other than that is false. Symbols p and q can represent any statement. Compound statement p and q depend on the truth value of each p and q. For example conjunction p ^ q wrong when p false and q true. Here is the truth table conjunction p ^ q more.

p	q	p^q
B	B	B
B	S	S
S	B	S
S	S	S

Truth table of Negation

Statement of negation is a statement which denied earlier statements, or opponents of the preliminary statement. To make the truth table of the negation statement, first we create the truth table for the truth value of statements of origin. The statement of origin (p) can be valuable right or wrong. So the truth value table of the statement p is:

p

B

S

If p is true, then ~ p would be worth one, because ~ p deny the truth of p. If p is wrong, ~ p is true. Truth table for negation describe the truth value given by ~ p. The first line in the truth table read "~ p false when p is true", while the second line read "~ p is true when p is wrong."

p	~p
B	S
S	B

Learning Truth Value

Assuming you have a friend named Burhan. He's a doctor, and you also know he's PDI-P cadre. If someone tells you, "Burhan a doctor and the Golkar Party cadres." You will say that the statement is false. On the other hand, to say to you, "Burhan a doctor or a cadre of the Golkar Party." Then you will justify that statement. These sentences is a compound sentence - a mix of simple sentences with conjunctions logic. Actually, when a statement could be true? Also when we say, wrong? To answer this question, we need to know the truth value of simple sentences that constitute them. After that the truth value of compound sentences can be in the know of the truth value of simple sentences that constitute them and how they are combined.

The value of truth of a statement is a classification of whether true or false statement, which can be denoted by B or S. In electronic discussion, especially the digital gate, the notation used is 1 to B and 0 to S. Examples of statements that is true is the 'Hyderabad is the capital of Central Java province. " So that statement is worth T. Certainly the sentence which says that the FC is the capital of Central Java province is the wrong phrase, valued F.

A simple way is used to determine the truth value of a compound sentence is to make a truth table. Table truth make all possible combinations of truth value of a simple statement of truth value is given. Truth table also provides the difference argument is valid and not valid.

Implication, or a conditional

Notice the phrase 'If it's raining, then the road became wet'. That phrase is a compound sentence, because it is formed by two statements, namely 'Now the rain' and 'roads become wet'. Note that the second sentence is linked by connective sentence 'If - Then -'. Any sentence of the form 'If p then q' is called a conditional (or implications), p is called the conditional hypothesis, q is called the conclusion of a conditional. Conditional 'if p then q', (consider if the same as if), symbolized by a p q (p implies q) . In everyday life, then the conditional word is often not mentioned. For example, if it's raining, the road became wet. Another way to write conditional is q if p (roads became wet when it's raining).

EXAMPLE

Using a symbolic representation

p: I'm healthy

q: I eat fast food.

r: I exercise regularly.

Express sentence under the representation simbolils.

a. I'm healthy when exercising regularly.

b. If I only eat fast food and do not exercise regularly, I was not feeling well.

COMPLETION

a. I'm healthy when exercising regularly is the conditional sentence "if - then - 'and can be represented again become

'if you exercise regularly, I am healthy'

then, this compound sentence is represented symbolically by r

b. If I only eat fast food and not exercising regularly, I was not feeling well. This sentence is conditional 'if - then -', but involving konsungsi and two negation in it. The symbolic representations of the above sentence is (q ^ ~ p) r.

EXAMPLE

Express sentence 'All men must die'

COMPLETION

sentence 'All human must die' can be denoted again if he is human, would die. Then, we define two symbolic sentence:

p: Something is a human

q: Human must die

sentence can be expressed p -> q. In general, the phrase 'all p then q' can be expressed p -> q

We have discussed how the compound statements are formed, the next we can give the truth value analysis.

compound statement	Symbols	Readings
Negation	~	notes
Conjunctions	^	and
disjunction	v	or
conditional / implications		if - then -

Disjunction, union and the sample space

Disjunction is formed when a few sentences connected by the word 'or'. Symbols used by the disjunction to represent the word 'or' is v. Thus, the disjunction of using 'pvq' to represent the sentence 'p or q'. We can interpret the word 'or' in two ways. Consider the following sentence:

p: Ahmad is a cadre PDIP
q: Syu'aib a cadre PDIP

Sentence Ahmad is a cadre of PDIP or Syu'aib is a cadre of PDIP may symbolized by p v q. Note that a possibility they can be a cadre PDIP together. In this example, we discuss about 'or' inclusive. Now consider the example

p: Ahmad is a cadre PDIP
q: Ahmad is a cadre of Democrats

In fact in our political world, it is very likely one man has two membership cadre. However, we assume each man only wanted to be a cadre in one party only. Thus, the two events are not likely to occur together. Technically, if the second sentence is true, then the combination of both false disjunction. In short one sentence mengexclude another sentence. Is called the disjunction Exclusive disjunction.

Conjunction, an intersection state

Consider the statement 'Abbas is a member of PMI and he Golkar Party cadres'. It is a compound sentence, since this sentence comes from two simple sentences - 'Abbas, a member of the Red Cross' and sentence 'He (Abbas) is the Golkar Party cadres' - and the conjunctive 'and'. Compound sentences like this are called conjunctions. Conjunction with two or more statements that are connected by conjunctions and. We use the symbol ^ to represent the word and, then, conjunctions 'p ^ q' represents a compound sentence 'p and q'.

EXAMPLE
Use a symbolic representation
p: Abbas, a member of PMI
q: Abbas is a Golkar cadre
Express the following compound statement using symbols:
a. Abbas is a member of the Red Cross and the Golkar Party cadres
b. Abbas, a member of PMI and he's not the Golkar Party cadres

COMPLETION
a. p ^ q
b. p ^ ~ q

Negation, Denial of Statement

Negation of a statement is a denial statement. Negation is represented using the symbol ~. Negation is often formed by adding the word 'no' in a sentence of origin. Examples are given the sentence "p: rainy day ', its negation is" ~ p: no rain today. " If the original sentence is true, its negation is false, and vice versa. Sentence truth value of negation is always contrary to the truth value of sentence of origin. Because the truth of a negation depends on the original sentence, the negation of a sentence kalmat classified as compound.

EXAMPLE
Write the sentence that represents the negation of each of the following sentences: a. Legislative candidate from Golkar Party, b. Legislative candidate was not from the Golkar Party. c. Several candidates from the Golkar party, d. All the candidates come from the democrat party, e. None of the party's legislative Hanura.

COMPLETION
a. Negation of 'legislative candidates from the Golkar Party' is 'is not a legislative candidate from Golkar Party'.
b. Negation of 'it is not a legislative candidate from Golkar Party' is 'legislative candidates from the Golkar Party'.
c. Errors often arise when asked the negation of 'a few candidates from the Golkar Party'. Many who answered 'some candidates did not come from the Golkar Party'. The statement did not reject the initial statement. The expression 'a few candidates from the Golkar Party' implies there is at least one candidate from the Golkar Party. Statements that deny this, of course, 'There are no candidates from the Golkar Party'. The statement is a negation of the initial statement.
d. Negation of 'all the candidates from the Golkar Party' is 'there is at least one candidate who did not come from the Golkar Party,' or phrase that is often used 'some candidates did not come from the Golkar Party'.
e. Negation 'No legislature of Hanura' is 'there is at least one person from the party's legislative Hanura'. When more than one interpretation at least, we can use another negation of 'some of the party's legislative Hanura.

Said some, all, and no known as quantifier, kuantor. Statement in c through e involve kuantor in it. Negation of 'all p is q' is 'some p not q'. And vice versa. And the negation 'some p is q' is 'does not exist p is q'.