Write each of the implications `(p implies q)` in the form `(~p vv q)` and hence write the negation of each statement:
If `5 gt7,` then `6 lt 4.`

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the statements p and q
Let:
- \( p \): "5 is greater than 7" (which is false)
- \( q \): "6 is less than 4" (which is also false)

### Step 2: Write the implication \( p \implies q \)
The implication \( p \implies q \) can be expressed in logical terms as:
\[ p \implies q \]

### Step 3: Rewrite the implication in the form \( \neg p \lor q \)
According to the logical equivalence, we can rewrite \( p \implies q \) as:
\[ \neg p \lor q \]

### Step 4: Determine \( \neg p \) and \( \neg q \)
Now, we need to find the negations of \( p \) and \( q \):
- \( \neg p \): "5 is not greater than 7" (which is true)
- \( \neg q \): "6 is not less than 4" (which is true)

### Step 5: Substitute \( \neg p \) and \( q \) into the expression
Now we substitute \( \neg p \) and \( q \) into the expression:
\[ \neg p \lor q = \text{"5 is not greater than 7"} \lor \text{"6 is less than 4"} \]

### Step 6: Write the negation of the implication
The negation of the implication \( p \implies q \) can be expressed as:
\[ \neg(p \implies q) \]
Using the equivalence we derived earlier, this can be rewritten as:
\[ p \land \neg q \]

### Step 7: Substitute \( p \) and \( \neg q \)
Now we substitute \( p \) and \( \neg q \):
- \( \neg q \): "6 is not less than 4" (which is true, meaning "6 is greater than or equal to 4")

Thus, we have:
\[ p \land \neg q = \text{"5 is greater than 7"} \land \text{"6 is greater than or equal to 4"} \]

### Final Statement
The negation of the statement "If 5 is greater than 7, then 6 is less than 4" is:
"5 is greater than 7 and 6 is greater than or equal to 4."

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