Membuktikan A ∩ B=B ∩ A

Show that:

1. A ∩ B=B ∩ A
2. (A ∩ B) ∩ C=A ∩ (B ∩ C)

Answer:
1. Proof:
Show that A ∩ B ⊂ B ∩ A
take any x ∈ (A ∩ B)
obvious x ∈ (A ∩ B)
⇔ x ∈ A ∧ x ∈ B
⇔ x ∈ B ∧ x ∈ A
⇔ x ∈ (B ∩ A)
so A ∩ B ⊂ B ∩ A...........( I )

Show that B ∩ A ⊂ A ∩ B
take any x ∈ (B ∩ A)
obvious x ∈ (B ∩ A)
⇔ x ∈ B ∧ x ∈ A
⇔ x ∈ A ∧ x ∈ B (komutatif)
⇔ x ∈ (A ∩ B)
so B ∩ A ⊂ A ∩ B.............( II )
From (!) and (II) we conclude that A ∩ B ⊂ B ∩ A


2. Proof :
Show that (A ∩ B) ∩ C ⊂ A ∩ (B ∩ C)
take any x ∈ [(A ∩ B) ∩ C]
obvious x ∈ [(A ∩ B) ∩ C]
⇔ x ∈ (A ∩ B) ∧ x ∈ C
⇔ (x ∈ A ∧ x ∈ B) ∧ x ∈ C
⇔ x ∈ A ∧ (x ∈ B ∧ x ∈ C)
⇔ x ∈ A ∧ x ∈ (B ∩ C)
⇔A ∩ (B ∩ C)
so (A ∩ B) ∩ C ⊂ A ∩ (B ∩ C)..............(I)

Show that A ∩ (B ∩ C) ⊂ (A ∩ B) ∩ C
take any x ∈ [A ∩ (B ∩ C)]
obvious x ∈ [A ∩ (B ∩ C)]
⇔ x ∈ A x ∈ (B ∩ C)
⇔ x ∈ A ∧ (x ∈ B ∧ x ∈ C)
⇔ (x ∈ A ∧ x ∈ B) ∧ x ∈ C
⇔ x ∈ (A ∩ B) ∧ x ∈ C
⇔ (A ∩ B) ∩ C
so A ∩ (B ∩ C) ⊂ (A ∩ B) ∩ C............(II)
From (I) and (!!) we conclude that (A ∩ B) ∩ C ⊂ A ∩ (B ∩ C)

Posted in: matematika