To demonstrate that the operators '-' (negation, $\neg$) and 'V' (disjunction, $\lor$) form a functionally complete collection of logical operators, we need to show that any logical expression can be constructed using only these two operators.

A functionally complete set of logical operators can be used to express all possible truth functions. In other words, any logical operation (AND, OR, NOT, IMPLICATION, etc.) can be expressed using only the operators in this set.

We will show how to express the basic logical operations AND ($\land$) and IMPLICATION ($\rightarrow$) using only $\neg$ and $\lor$.

1. Negation ($\neg$)

The negation operator $\neg$ is already part of the set.

2. Disjunction ($\lor$)

The disjunction operator $\lor$ is already part of the set.

3. Conjunction ($\land$)

To express conjunction using only $\neg$ and $\lor$, we use De Morgan's laws:

$A \land B \equiv \neg (\neg A \lor \neg B)$

This expresses $A \land B$ using only $\neg$ and $\lor$.

4. Implication ($\rightarrow$)

To express implication using only $\neg$ and $\lor$, we use the definition of implication:

$A \rightarrow B \equiv \neg A \lor B$

This expresses $A \rightarrow B$ using only $\neg$ and $\lor$.

5. Biconditional ($\leftrightarrow$)

To express biconditional using only $\neg$ and $\lor$, we can use the equivalence of biconditional:

$A \leftrightarrow B \equiv (A \rightarrow B) \land (B \rightarrow A)$ $\equiv (\neg A \lor B) \land (\neg B \lor A)$ $\equiv \neg (\neg (\neg A \lor B) \lor \neg (\neg B \lor A))$ $\equiv \neg ((A \land \neg B) \lor (B \land \neg A))$ $\equiv \neg ((\neg B \lor A) \land (\neg A \lor B))$

6. Exclusive OR ($\oplus$)

To express exclusive OR using only $\neg$ and $\lor$:

$A \oplus B \equiv (A \land \neg B) \lor (\neg A \land B)$ $\equiv \neg (\neg A \lor B) \lor \neg (A \lor \neg B)$

This expresses $A \oplus B$ using only $\neg$ and $\lor$.

Conclusion

We have shown how to express AND ($\land$), OR ($\lor$), NOT ($\neg$), IMPLICATION ($\rightarrow$), BICONDITIONAL ($\leftrightarrow$), and XOR ($\oplus$) using only the operators $\neg$ and $\lor$. Since these operations are sufficient to express any logical formula, we have demonstrated that the set $\{ \neg, \lor \}$ is functionally complete.

Do you have any questions or need further details?

Here are 5 more questions you might be interested in:

1. What are the De Morgan's laws and how are they used in logic?
2. How can you prove that NAND and NOR are functionally complete sets of operators?
3. What is the significance of functional completeness in logic?
4. Can you explain the process of constructing a truth table for logical expressions?
5. How does functional completeness relate to digital circuit design?

Tip: Understanding functional completeness can greatly simplify the design of logical circuits and the minimization of logical expressions, making it a powerful concept in both theoretical and applied logic.