Logika, Proporsi dan Komponen Penghubungnya

Pengertian Logika, Proporsi, dan Komponen Penghubungnya

Materi Kuliah Pengantar Dasar Matematika

by Ardhi Prabowo

Universitas Negeri Semarang

1. Logika dan Kalimat Berarti 

Agar komunikasi dapat dimengerti digunakan logika sebagai kontrol. Dalam matematika, bahasa komunikasinya disebut kalimat matematika yaitu kalimat yang menggunakan lambang-lambang matematika.

Kalimat berarti terbagi atas: Kalimat Pernyataan dan Bukan Pernyataan (Kalimat Terbuka)

Contoh:

(silahkan sebutkan yang contoh dan yang tak contoh)

Jadi apakah yang disebut dengan pernyataan (proporsi)?

        Kalimat Terbuka

  Menurut Saudara, apa yang dimaksud dengan kalimat terbuka?

  A simple statement is one that does not contain any other statement as a part. We will use the lower-case letters, p, q, r, as symbols for simple statements. In Indonesian, we say it Proposisi.

  A compound statement is one with two or more simple statements as parts or what we will call components. A component of a compound is any whole statement that is part of a larger statement; components may themselves be compounds. In Indonesian, we usually say Komposit or Proposisi Komposit.

            Tugas 1

Buatlah 5 Contoh masing-masing kalimat dibawah ini:

1.      Kalimat pernyataan

2.      Kalimat terbuka

3.      Kalimat perintah

4.      Kalimat tanya

5.      Kalimat harapan

6.      Kalimat faktual.

Kemudian diskusikan dengan teman anda kalimat yang telah anda buat.

 

            Tugas 2

Tentukanlah kalimat berikut ini merupakan pernyataan atau bukan, dan jika merupakan kalimat terbuka tentukanlah variabelnya!

1)      Semarang terletak di Jawa Tengah.

2)      Terdapat sebuah bilangan prima yang genap.

3)      5x  + 8x = 12x

4)      cos 2x = cos²x – sin²x.

5)      Semua siswa mengerjakan soal-soal latihan 1.

6)      Mudah-mudahan semua siswa kelas II naik kelas.

7)      7x – 8 > 6 + 3x

8)      Mengapa kamu  terlambat datang ke sekolah?

 

Operator of Statements

• An operator (or connective) joins simple statements into compounds, and joins compounds into larger compounds. We will use the symbols Ù , Ú , Þ , Û and to designate the sentential connectives. They are called sentential connectives because they join sentences (or what we are calling statements). The symbol, ~ , is the only operator that is not a connective; it affects single statements only, and does not join statements into compounds.

• Special for symbol º , we usually called equivalent, is used to explain the similarity of two compounds.

 

Truth Value and Truth-Functional

• The truth value of a statement is its truth or falsity. All meaningful statements have truth values, whether they are simple or compound, asserted or negated. That is, p is either true or false, ~p is either true or false, p Ú q is either true or false, and so on.

• A compound statement is truth-functional if its truth value as a whole can be figured out solely on the basis of the truth values of its parts or components. A connective is truth- functional if it makes only compounds that are truth-functional. For example, if we knew the truth values of p and of q, then we could figure out the truth value of the compound, p Ú q. Therefore the compound, p Ú q, is a truth-functional compound and disjunction is a truth-functional connective. 

•  All four of the connectives we are studying (disjunction, conjunction, implication, and equivalence) are truth-functional. Negation is a truth-functional operator. With these four connectives and negation we can express all the truth-functional relations among statements. (Can you imagine how we would prove this?)