TUGAS 4 PDM

Show that :

1. 
   
   A ∩ B = B ∩ A
   

1. 
   
   Show that A ∩ B ⊂ B ∩ A
   
   
   

Take any x ∈ A ∩ B

Obvious x ∈ A ∩ B

ek x ∈ A ∧ x ∈ B

ek x ∈ B ∧ x ∈ A

ek x ∈ B ∩ A

So A ∩ B ⊂ B ∩ A

2. 
   
   Show that B ∩ A ⊂ A ∩ B
   

Take any x ∈ B ∩ A

Obvious x ∈ B ∩ A

ek x ∈ B ∧ x ∈ A

ek x ∈ A ∧ x ∈ B

ek x ∈ A ∩ B

So B ∩ A ⊂ A ∩ B

From (i) and (ii) we conclude that A ∩ B = B ∩ A

2. 
   
   ( 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

ek x ∈ ( A ∩ B ) ∧ x ∈ C

ek ( x ∈ A ∧ x ∈ B ) ∧ x ∈ C

ek x ∈ A ∧ ( x ∈ B ∧ x ∈ C )

ek x ∈ A ∩ ( B ∩ C )

So ( A ∩ B ) ∩ C ⊂ A ∩ ( B ∩ C )

2. 
   
   Show that A ∩ ( B ∩ C ) ⊂ ( A ∩ B ) ∩ C
   

Take any x ∈ A ∩ ( B ∩ C )

Obvious x ∈ A ∩ ( B∩ C )

ek x ∈ A ∧ x ∈ ( B ∩ C )

ek x ∈ A ∧ ( x ∈ B ∧ x ∈ C )

ek ( x ∈ A ∧ x ∈ B ) ∧ x ∈ C

ek x ∈ ( A ∩ B ) ∧ x ∈ C

So A ∩ ( B ∩ C ) ⊂ ( A ∩ B ) ∩ C

From (i) and (ii) we conclude that ( A ∩ B ) ∩ C = A ∩ ( B ∩ C )