linearizing if-then with two binary invoved

Question

I need to linearize this $\alpha_i^{ap}=1 \implies \beta_{ij}^{ap} =1 $. As both are binary, I think the right answer is $ \alpha_i^{ap} \ge \beta_{ij}^{ap} $. But rading one of the questions a the answer by @Erwin Kalvelagen makes me to wonder whether that's really correct. When I apply that method it doesn't seem to be correct either. Whcih one is correct?

Answer 1

You have the inequality backwards. It should instead be $\alpha_i^{ap} \le \beta_{ij}^{ap}$. Here's a derivation using conjunctive normal form: $$ x \implies y \\ \lnot x \lor y \\ (1 - x) + y \ge 1 \\ x \le y $$