Sunday, January 22, 2012
Proof Ex 4.17.1 Subset
Show that A ⊄ B ⇔ A - B ≠ ∅
Proof:
A ⊄ B
= ¬∀x(x ∈ A ⇒ x ∈ B)
= ¬∀x(x ∉ A ∨ x ∈ B)
= ∃x(¬(x ∉ A ∨ x ∈ B))
= ∃x(x ∈ A ∧ x ∉ B)
⇔ A - B ≠ ∅
Show that A ⊄ B ⇔ A - B ≠ ∅
Proof:
A ⊄ B
= ¬∀x(x ∈ A ⇒ x ∈ B)
= ¬∀x(x ∉ A ∨ x ∈ B)
= ∃x(¬(x ∉ A ∨ x ∈ B))
= ∃x(x ∈ A ∧ x ∉ B)
⇔ A - B ≠ ∅