Bianca's seeming innocence and obedience at the play's beginning works as a foil to Kate's shrewish behavior. In fact, when we first see the women, Tranio and Lucentio point out that they are total and complete opposites – Kate acts like a crazy, loud-mouthed wench and Bianca behaves like an obedient and "silent" daughter. Turns out, though, that Bianca isn't as sweet or silent as she first appears. She tricks her dad, screws around with her suitors, talks dirty to her brother-in-law, and treats her husband like a chump at her wedding reception. By the play's end, it is Bianca, not Kate, who acts like a shrew.
We know what you're thinking. How the heck can Petruchio be a protagonist and a foil? We're not trying to make your life a nightmare, honestly. We just want to show you how these devices can be complex and fun. Here's how it works: Petruchio becomes a "foil" to Kate's character when he begins his "shrew-taming" campaign. He mirrors Kate's shrewish behavior when he purposefully behaves like an abusive husband and master so that Kate will get a dose of her own medicine. This is especially clear when Petruchio yells at his servants and kicks them around for no good reason. (Check out Act 4, Scene 1, where Petruchio and Kate arrive at Petruchio's house after their wedding.) These scenes echo earlier moments where Kate threatens to bloody Hortensio's nose and ties up her sister. In fact, one of Petruchio's servants, Curtis, remarks that Petruchio is "more shrew than" Kate (4.1.15).
Both Sly and Katherine are seen by other characters as unruly and both are punished for their bad behavior by being publicly humiliated. Sly is "punked" in an elaborate scheme to make him believe he is a nobleman and Kate is humiliated by Petruchio from the moment she meets him. Mostly, she's humiliated when Petruchio arrives late to the wedding and proceeds to act outrageously. He also forces Kate to play along with his crazy antics throughout the text.
This brings us to our next point. Both Sly and Kate are forced to participate in a charade of some sort. When Sly is tricked into believing he is a Lord, he's inserted into a kind of play without knowing it. (Sort of like Jim Carey's character in the movie The Truman Show.) Everyone around Sly is playing a role and they basically force Sly to play the part of a nobleman. Kate, as we've discussed before, is forced by Petruchio to play-act as well. She has to go along with Petruchio when he pretends an old man is a young virgin and also when he pretends the sun is the moon. She may also be acting when she delivers her final speech of the play, a laundry list of all the reasons why women should be obedient to men. In other words, she's forced to play the role of obedient wife. It's easy to forget about Sly, especially since he disappears from the play after the end of act one, scene one. Still, it can be useful to rethink his role in the Induction after you finish reading the five-act inset play. In fact, it's really useful to rethink the entire Induction as a kind of foil to the inset play.