It's a little confusing at first. It somewhat depends on the event.

Elite Guard and Gambit: Just having the cards does the trick

PeroColo: The card must be in your EVENT deck, which is totally separate from your My Pero deck in the cards tab of the game.

Also, "Slayer" cards are Special Effect Cards, which increase the rate at which you do something in the event. (Increase SED/Collection of trinkets)

N-Slayers are now won through the N-Gacha.
R Slayers and SR Slayers are won through the E-Gacha.

As someone mentioned, there are two phases to an event, so the cards released in the second phase will be the Alphas, and the previous Alphas will be Betas.