Each of the following open senteces defines a relation R in the natural number N. State whether each relation is transitive or not. b. "x is divisible by y"

Pertanyaan

Each of the following open senteces defines a relation $R$ in the natural number $N$ . State whether each relation is transitive or not.

b. " $x$ is divisible by $y$ "

I. Kumaralalita

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

Jawaban terverifikasi

Jawaban

relasi

merupakan relasi transitif.

Pembahasan

Relasi R space colon space A rightwards arrow A disebut relasi transitif jika untuk setiap left parenthesis a comma space b right parenthesis element of R dan left parenthesis b comma space c right parenthesis element of R , maka berlaku .

Diberikan kalimat terbuka " habis dibagi " yang mendefinisikan relasi dalam bilangan asli . Relasi tersebut dapat dituliskan dalam himpunan pasangan berurutan :

R equals open curly brackets left parenthesis 2 comma 1 right parenthesis comma space left parenthesis 4 comma 2 right parenthesis comma space left parenthesis 4 comma 1 right parenthesis comma space left parenthesis 12 comma 6 right parenthesis comma space left parenthesis 6 comma 3 right parenthesis comma space left parenthesis 12 comma 3 right parenthesis comma space... close curly brackets .

Apabila diambil a comma space b comma space c element of A , jika habis dibagi dan habis dibagi , maka juga habis dibagi . Sehingga relasi memenuhi untuk setiap left parenthesis a comma space b right parenthesis element of R dan left parenthesis b comma space c right parenthesis element of R , berlaku .

Jadi, relasi merupakan relasi transitif.

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