# pseudo dimension of the minimum of functions

Suppose a real-valued function class $$\mathcal{F}$$ with pseudo dimension less than $$d$$, I am wondering what is the pseudo dimension of the following function class $$$$\mathcal{F}_2 = \{\min(f_1, f_2), \text{with } f_1,f_2\in \mathcal{F}\}$$$$ where for any $$x$$, $$\min(f_1, f_2)(x) = \min(f_1(x), f_2(x))$$. My intuition tells me that the pseudo dimension of $$\mathcal{F}_2$$ should be bounded by $$2d$$, but I just cannot prove it.

Any help is appreciated!

• Can you elaborate your intuition? The more, the better. If you have tried proving, can you tell where lies the difficulty? May 9 at 4:55