Forgive and Forget? No.

Merriam Webster defines "forgive" as a transitive verb. "to cease to feel resentment against (an offender) : PARDON" i.e. "forgive one's enemies"

The definition of forgive says nothing about forgetting the offense. Nor does it suggest that you do it to benefit the other person. To forgive frees you from resentment and anger. It benefits you to make you life easier, better, freer.

Forgetting the offense would not be useful. If you forgot it happened, how would you prevent it happening again in the future? Every experience gives us information to us in determining how to make your way with people, how to evaluate situations, etc. No, don't forget.

But do let go of the feelings that may still linger when you remember the event. Forgiveness enables you to clear the slate with people and move on. 

When a community, or group, or country, etc., harbors resentment against another, there is very little trust that can be built. And behavior choices are more likely to be made on the basis of feelings than on fact. How many wars and acts of violence between groups would be avoided if true forgiveness had occurred? 

Forgive others, whether they know it or not. Forgive others, whether they even care or not. Do it for your betterment and to make the world better.

Resolve conflicts peacefully and learn from the experience. Keep your old emotions out of future situations. Let the resentment go.

Forgive.