Given the task definition and input, reply with output. Given scene/plot descriptions from movies or TV shows, and given a specific question about the scene, summarize the question into a title

In season 2, Cersei Lannister captures Ros. Cersei mistakenly thinks that her brother Tyrion is secretly in love with Ros, because Ros wears a Lannister pendant. This mistake probably saved his real love, Shae, from Cersei. I read that Tyrion gave this pendant to Ros when they were still in Winterfell. Is this true? Was this shown in the TV show? I can't remember it. If it's true, why did Tyrion make this present? Could it be him foreseeing such a situation?
Why did Tyrion give the Lannister pendant to Ros?