HMMT November 2016 U13

How many functions $f:\{0,1\}^{3} \rightarrow\{0,1\}$ satisfy the property that, for all ordered triples $\left(a_{1}, a_{2}, a_{3}\right)$ and $\left(b_{1}, b_{2}, b_{3}\right)$ such that $a_{i} \geq b_{i}$ for all $i, f\left(a_{1}, a_{2}, a_{3}\right) \geq f\left(b_{1}, b_{2}, b_{3}\right)$ ?