Explanation:
The first law of thermodynamics is also known as the law of conservation of energy. It states that energy cannot be created or destroyed, only transferred or converted from one form to another. In other words, the total energy of a system and its surroundings remains constant.

Mathematical representation:
The first law of thermodynamics can be mathematically represented using the equation:

ΔU = Q - W

where ΔU is the change in internal energy of the system, Q is the heat transferred to the system, and W is the work done by the system.

Now let's break down the options given in the question:

a. ΔU = -Q + W
This equation is not correct because it implies that the internal energy of the system decreases when heat is added to it, which violates the first law of thermodynamics.

b. ΔU = Q + W
This equation is also not correct because it implies that the internal energy of the system increases when work is done on it, which again violates the first law of thermodynamics.

c. ΔU = Q - W
This equation is correct and represents the first law of thermodynamics. It states that the change in internal energy of the system is equal to the heat added to the system minus the work done by the system.

d. ΔU = -Q - W
This equation is not correct because it implies that the internal energy of the system decreases when both heat and work are added to it, which again violates the first law of thermodynamics.

Therefore, the correct answer is option 'C', which represents the first law of thermodynamics.