Now, let's look at the second term:

Here, implies that we are sampling a random noise from the uniform distribution . implies that the generator takes the random noise as an input and returns a fake image based on its implicitly learned distribution .

implies that we are feeding the fake image generated by the generator to the discriminator and it will return the probability of the fake input image being a real image.

If we subtract from 1, then it will return the probability of the fake input image being a fake image:

Since we know is not a real image, the discriminator will maximize this probability. That is, the discriminator maximizes the probability of being classified as a fake image, so we write:

Instead of maximizing raw probabilities, we maximize the log probability:

implies the expectations of the log likelihood of the input images generated by the generator being fake.