TUGAS PDM 4

Show that :
a) A ∩ A = A
b) A ∩ B = B ∩ A
c) ( A ∩ B ) ∩ C = A ∩ ( B ∩ C )

Answer

a) Proof :

i. Show that A ∩ A ⊂ A
Take any x ∈ A ∩ A
Obvious x ∈ A ∩ A
≡ x ∈ A ∧ x ∈ A
≡ x ∈ A (idempoten)
So, A ∩ A ⊂ A .................................(1)

ii. Show that A ⊂ A ∩ A
Take any x ∈ A
Obvious x ∈ A
≡ x ∈ A
≡ x ∈ A ∧ x ∈ A (idempoten)
So, A ⊂ A ∩ A .................................(2)

From (1) and (2), we conclude that A ∩ A = A


b) Proof :

i. Show that ( A ∩ B ) ⊂ ( B ∩ A )
Take any x ∈ A ∩ B
Obvious x ∈ A ∩ B
≡ x ∈ A ∧ x ∈ B
≡ x ∈ B ∧ x ∈ A (komutatif)
So, ( A ∩ B ) ⊂ ( B ∩ A )......................(1)

ii. Show that( B ∩ A ) ⊂ ( A ∩ B )
Take any x ∈ B ∩ A
Obvious x ∈ B ∩ A
≡ x ∈ B ∧ x ∈ A
≡ x ∈ A ∧ x ∈ B (komutatif)
So, ( B ∩ A ) ⊂ ( A ∩ B ) ......................(2)

From (1) and (2), we conclude that A ∩ B = B ∩ A

c) Proof :

i. Show that ( A ∩ B ) ∩ C = A ∩ ( B ∩ C )
Take any x ∈ ( A ∩ B ) ∩ C
Obvious x ∈ ( A ∩ B ) ∩ C
≡ (x ∈ A ∧ x ∈ B) ∧ x ∈ C
≡ x ∈ A ∧ ( x ∈ B ∧ x ∈ C ) (asosiatif)
So, [( A ∩ B ) ∩ C] ⊂ [A ∩ ( B ∩ C )]...........(1)

ii. Show that A ∩ ( B ∩ C ) = (A ∩ B ) ∩ C
Take any x ∈ A ∩ ( B ∩ C )
Obvious x ∈ A ∩ ( B ∩ C )
≡ x ∈ A ∧ ( x ∈ B ∧ x ∈ C)
≡ ( x ∈ A ∧ x ∈ B ) ∧ x ∈ C (asosiatif)
So, [A ∩ ( B ∩ C )] ⊂ [( A ∩ B ) ∩ C] ...........(2)

From (1) and (2), we conclude that ( A ∩ B ) ∩ C = A ∩ ( B ∩ C )