Which of the following is equivalent to ` p Rightarrow q` ?

Options

A. ` p Rightarrow q`
B. ` q Rightarrow p`
C. ` ( p Rightarrow q) ^^ ( q Rightarrow p)`
D. none of these

Answer

To determine which of the following is equivalent to \( p \Rightarrow q \) (read as "p implies q"), we can use the properties of logical implications.

### Step-by-Step Solution:

1. **Understanding Implication**: The statement \( p \Rightarrow q \) can be interpreted as "if p is true, then q is also true." This means that the only time \( p \Rightarrow q \) is false is when p is true and q is false.

2. **Using Logical Equivalence**: The implication \( p \Rightarrow q \) can be rewritten using logical operations. It is equivalent to \( \neg p \lor q \) (not p or q). This means that either p is false, or q is true (or both).

3. **Identifying Equivalent Expressions**:
- From the logical equivalence, we know that:
\[ p \Rightarrow q \equiv \neg p \lor q \]
- Additionally, we can also express \( p \Rightarrow q \) using the contrapositive, which states that \( \neg q \Rightarrow \neg p \) is also equivalent to \( p \Rightarrow q \).

4. **Conclusion**: Based on the above reasoning, any expression that can be simplified to \( \neg p \lor q \) or \( \neg q \Rightarrow \neg p \) will be equivalent to \( p \Rightarrow q \).

### Final Answer:
Thus, the expressions that are equivalent to \( p \Rightarrow q \) are:
- \( \neg p \lor q \)
- \( \neg q \Rightarrow \neg p \)