MathGhotic

1.)Let A , B set and x is an element . While :
a) x ∈ A ∩ B ⇔ x ∈ A ∧ x ∈ B
b) x ∉ A ∩ B ⇔ x ∉ A ∧ x ∉ B

Answer:
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.) 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) ....(i)
So A ∩ A ⊂A ⇔ x ∈ A
ii)Show that A ⊂A ∩ A
Take any x ∈ A ∩ A
Obvious x ∈ A ∩ A
⇔ x ∈ A ∧ x ∈ A
⇔ x ∈ A (idempoten) ....(ii)
So A ∩ A ⊂A ⇔ x ∈ A
From that (i) and (ii) 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 ∩ A
So A ∩ B ⊂ B ∩ A (komutatif )...(i)
ii) Show that B ∩ A ⊂A ∩ B
Take any x ∈ B ∩ A
Obvious x ∈ B ∩ A
⇔ x ∈ B ∧ x ∈ A
⇔ x ∈ A ∩ B
So B ∩ A ⊂ A ∩ B (komutatif )...(ii)
From that (i) and (ii) 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)
⇔ x ∈ A ∩ (B ∩ C )
So (A ∩ B) ∩ C ⊂ A ∩ (B ∩ C ) (asosiatif)...(i)
ii) i) Show tha) 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
⇔ x ∈ (A ∩ B ) ∩ C
So A ∩ (B ∩ C ) ⊂ (A ∩ B) ∩ C (asosiatif)...(ii)
From that (i) and (ii) we conclude that (A ∩ B) ∩ C = A ∩ (B ∩ C )

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