TUGAS 4

1. Let A,B set and x is an elements. While :
a. x ∈ A ∩ B
b. x∉ A ∩ B

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

The answer:
1. a). x ∈ A ∩ B ↔ x ∈ A ∧ x ∈ B
b). PBE :
1. x ∉ A ∩ B
2. Tidak benar bahwa x ∈ A ∩ B
3. Tidak benar bahwa x ∈ A ∧ x ∈ B
4. x ∉ A ∧ x ∉ B

2. a). Proof :
i) Show that A ∩ A ⊂ A
Take any x ∈ A ∩ A
Obvious x ∈ A ∩ A
↔ x ∈ A ∧ x ∈ A
↔ x ∈ A
So, A ∩ A ⊂ A
ii) Show that A ⊂ A ∩ A
Take any x ∈ A
Obvious x ∈ A
↔ x ∈ A ∧ x ∈ A
↔ x ∈ A
So, A ⊂ A ∩ A
From (i) and (ii) we conclude that A ∩ A = A

b). Proof :
i). Show that A ∩ B ⊂ B ∩ A
Take any x ∈ A ∩ B ∧ x∈ B∩ A
Obvious x ∈ A ∩ B ∧ x ∈ B ∩ A
↔ x ∈(A ∩ B) ∧ x ∈ (B ∩ A)
↔ x ∈ {(A ∩ B) ∧ (B ∩ A)}
So A ∩ B ⊂ B ∩ A

ii). Show that B ∩A ⊂ A ∩ B
Take any x ∈ B ∩ A ∧ x ∈ A ∩ B
Obvious x ∈ B ∩ A ∧ x ∈ A ∩ B
↔ x ∈ (B ∩ A) ∧ x ∈ (A ∩ B)
↔ x ∈ {(B ∩ A) ∧ (A ∩ B)}
So B ∩ A ⊂ A ∩ B
From (i) and (ii) we conclude that A ∩ B = B ∩ A (H.Komutatif)

c). Proof :
i) Show that (A ∩ B)∩ C ⊂ A ∩ (B ∩ C)
Take any x ∈ (A ∩ B) ∩ C ∧ x ∈ A ∩ (B ∩ C)
Obvious x ∈ (A ∩ B) ∩ C ∧ x ∈ A ∩(B ∩ C)
↔ x ∈ (A ∩ B) ∩ C ∧ x ∈ A ∩ (B ∩ C)
↔ x ∈ {(A ∩ B) ∩ C ∧ A ∩ (B ∩ C)}
So (A ∩ B) ∩ C ⊂ A ∩ (B ∩ C)

ii) Show that A ∩(B ∩ C) ⊂ (A ∩ B)∩ C
Take any x ∈ A ∩(B ∩ C) ∧ x ∈ (A ∩ B)∩ C
Obvious x ∈ A ∩(B ∩ C) ∧ x ∈(A ∩ B)∩ C
↔ x ∈ A ∩ (B ∩ C) ∧ x ∈ (A ∩ B) ∩ C
↔ x ∈ {A ∩ (B ∩ C) ∧ (A ∩ B) ∩ C}
So A ∩(B ∩ C) ⊂ (A ∩ B)∩ C
From (i) and (ii) we conclude that A ∩(B ∩ C) = (A ∩ B)∩ C