Sunday, January 22, 2012
Proof Ex 4.17.2 Intersection
Show that A ∩ B = A - (A - B)
Proof:
A - (A - B)
= x ∈ A ∧ x ∉ (A - B)
= x ∈ A ∧ ¬(x ∈ A ∧ x ∉ B)
= x ∈ A ∧ (x ∉ A ∨ x ∈ B)
= (x ∈ A ∧ x ∉ A) ∨ (x ∈ A ∧ x ∈ B)
= x ∈ A ∧ x ∈ B
⇔ A ∩ B
Show that A ∩ B = A - (A - B)
Proof:
A - (A - B)
= x ∈ A ∧ x ∉ (A - B)
= x ∈ A ∧ ¬(x ∈ A ∧ x ∉ B)
= x ∈ A ∧ (x ∉ A ∨ x ∈ B)
= (x ∈ A ∧ x ∉ A) ∨ (x ∈ A ∧ x ∈ B)
= x ∈ A ∧ x ∈ B
⇔ A ∩ B