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The image you've uploaded contains a discrete mathematics problem related to logic expressions. The problem asks which of the given logical statements are tautologies. A tautology is a statement that is always true, regardless of the truth values of its components.

The logical statements provided in the image are:

a) \( F = \neg (\neg A \land \neg B) \)
b) \( G = (\neg A \rightarrow \neg B) \lor (\neg B \rightarrow \neg A) \)
c) \( G = (\neg P \lor Q) \rightarrow P \)
d) \( G = \neg \neg (\neg A \lor \neg B) \)
e) \( G = (\neg P \rightarrow Q) \lor (P \rightarrow Q) \)

To determine if these statements are tautologies, let's analyze them one by one.

### Analysis:

**a) \( F = \neg (\neg A \land \neg B) \):**

This can be rewritten using De Morgan's law:

\[
F = A \lor B
\]

This is not a tautology since the expression \( A \lor B \) is not always true; it depends on the truth values of \( A \) and \( B \).

**b) \( G = (\neg A \rightarrow \neg B) \lor (\neg B \rightarrow \neg A) \):**

This is logically equivalent to:

\[
G = (A \lor B) \land (\neg A \lor \neg B)
\]

This is the expression for the exclusive OR (XOR). XOR is not a tautology because it depends on the truth values of \( A \) and \( B \).

**c) \( G = (\neg P \lor Q) \rightarrow P \):**

This expression can be checked with truth tables or logical equivalences, but it is generally not a tautology since it depends on the values of \( P \) and \( Q \).

**d) \( G = \neg \neg (\neg A \lor \neg B) \):**

This simplifies to:

\[
G = \neg A \lor \neg B
\]

This is again not a tautology, as it is only true when \( A \) and \( B \) are not both true.

**e) \( G = (\neg P \rightarrow Q) \lor (P \rightarrow Q) \):**

This can be rewritten using logical equivalences:

\[
G = (P \lor \neg P) \rightarrow Q
\]

Since \( P \lor \neg P \) is a tautology (it is always true), this expression simplifies to:

\[
G = Q \rightarrow Q
\]

This is a tautology because \( Q \rightarrow Q \) is always true regardless of the truth value of \( Q \).

### Conclusion:

The correct answer is **e)** \( G = (\neg P \rightarrow Q) \lor (P \rightarrow Q) \) because it is a tautology.

Would you like further details or have any questions? Here are five related questions to deepen your understanding:

1. What is the difference between a tautology and a contradiction?
2. How does De Morgan's law help in simplifying logical expressions?
3. What are the truth tables, and how are they used to determine tautologies?
4. Can you explain the concept of logical equivalence with examples?
5. How does XOR function differ from OR and AND in logic?

**Tip:** When analyzing logical statements, it's often useful to simplify expressions using logical equivalences and to check for tautologies using truth tables.

Explore a discrete mathematics problem on logical statements and tautologies. Learn to analyze logical expressions and determine tautologies using De Morgan's Law and logical equivalences. Suitable for advanced learners interested in discrete mathematics.

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