Predicate Calculus Recurring Expressions

• All A is B:
  $\forall x(A(x) \rightarrow B(x))$
• No A is B:
  $\forall x (A(x) \rightarrow \neg B(x))$
• Some A is B:
  $\exists x(A(x) \wedge B(x))$
• Some A is not B:
  $\exists(A(x) \wedge \neg B(x))$
• All and only A is B:
  $\forall x(A9X) \equiv B(x)$
• Only A is B:
  $\forall x (B(x) \rightarrow A(x))$
• Not all A is B:
  $\exists x (A(x) \wedge \neg B(x))$
• All A is not B:
  $\forall x (A(x) \rightarrow \neg B(x))$