Write the negation of the following statement in terms of symbols :
A necessary condition that two triangles are equivalent is that they have the same area.

Answer

To write the negation of the statement "A necessary condition that two triangles are equivalent is that they have the same area" in terms of symbols, we can follow these steps:

### Step 1: Identify the components of the statement
We need to express the original statement in the form of a logical implication. Here, we can define:
- Let \( P \): "Two triangles are equivalent."
- Let \( Q \): "They have the same area."

The original statement can be expressed as:
\[ P \implies Q \]
This means "If two triangles are equivalent, then they have the same area."

### Step 2: Write the negation of the implication
The negation of an implication \( P \implies Q \) can be expressed using the logical equivalence:
\[ \neg(P \implies Q) \equiv P \land \neg Q \]
This means that the negation of "If P then Q" is equivalent to saying "P is true and Q is false."

### Step 3: Substitute back the definitions of P and Q
Now we substitute back the definitions of \( P \) and \( Q \):
- \( P \): "Two triangles are equivalent."
- \( \neg Q \): "They do not have the same area."

Thus, the negation of the original statement becomes:
\[ P \land \neg Q \]
Which translates to:
"Two triangles are equivalent and they do not have the same area."

### Final Answer
The negation of the statement "A necessary condition that two triangles are equivalent is that they have the same area" in terms of symbols is:
\[ P \land \neg Q \]
or in words:
"Two triangles are equivalent and they do not have the same area."

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