TUGAS 4 PDM

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 (komutatif)
≡ x ∈ (B ∩ A)
so A ∩ B ⊂ B ∩ A.........................( 1 )

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..........................( 2 )
From (1) and (2) 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) (assosiatif)
≡ x ∈ A ∧ x ∈ (B ∩ C)
≡ A ∩ (B ∩ C)
so (A ∩ B) ∩ C ⊂ A ∩ (B ∩ C).....................(1)

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 (assosiatif)
≡ x ∈ (A ∩ B) ∧ x ∈ C
≡ (A ∩ B) ∩ C
so A ∩ (B ∩ C) ⊂ (A ∩ B) ∩ C...................... (2)
From (1) and (2) we conclude that (A ∩ B) ∩ C ⊂ A ∩ (B ∩ C)