Which of the following statements is incorrect? (1) As the temperature of a gas increases, its volume increases. (2) As the temperature of a gas increases, its density decreases. (3) As the temperature of a gas increases, its density increases. (4) \(\operatorname{Both}(1)\) and \((3)\)

In statement (1), it says "As the temperature of a gas increases, its volume increases." This statement describes the behavior of gases according to Charles's Law, which states that for a constant pressure, the volume of a gas is directly proportional to its temperature (in Kelvin). Thus, as the temperature increases, the volume increases too. Therefore, statement (1) is correct.

Statement (2) states that "As the temperature of a gas increases, its density decreases." Considering the Ideal Gas Law, given by \(PV=nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the amount of gas in moles, \(R\) is the ideal gas constant, and \(T\) is temperature. Density, \(\rho\), is defined as the ratio of mass, \(m\), to volume, \(V\), or \(\rho = \frac{m}{V}\). We can obtain the relationship between density, temperature, and pressure from the Ideal Gas Law. Since \(n=\frac{m}{M}\), where \(M\) is the molar mass, we can rewrite the Ideal Gas Law as \(PV=\frac{m}{M}RT\). Rearranging, we get \(\rho=\frac{m}{V}=\frac{PM}{RT}\). Since \(P\), \(M\), and \(R\) are constants in this scenario, we can say that as the temperature, \(T\), increases, the density, \(\rho\) indeed decreases. Therefore, statement (2) is correct.

Statement (3) states that "As the temperature of a gas increases, its density increases." This contradicts our analysis in Step 2 and the relationship between temperature and density, as we found that as temperature increases, density decreases. Thus, statement (3) is incorrect.

We found that statement (3) is incorrect. Now, we need to verify if both statements (1) and (3) are incorrect, as claimed by statement (4). We found in Step 1 that statement (1) is correct, so statement (4) is also incorrect since it's saying both (1) and (3) are incorrect.

After analyzing each statement and comparing them with the Gas Laws, we can conclude that statement (3) - "As the temperature of a gas increases, its density increases" - is incorrect.