In Shakespeare's comedy, Taming of the Shrew, Katherine and Bianca are different in their own ways. Katherine is the protagonist of the play and the shrew referenced in the title. She is the daughter of Baptist and the older sister of Bianca. She has no suitors because of her plain appearance and shrewish nature. It's obvious that Katherine's father, Baptist, hasn't treated her as well as he treats Bianca. In contrast to Katherine, Bianca is the sweet and submissive daughter. Bianca is Katherine's younger sister. She has many suitors due to her attractive appearance and personality. She is her father's favorite and liked by everyone for her kind behavior.
From the beginning of the play, Katherine Minola appears to be what many people would call a shrew. A shrew is not just a talkative woman; she is a woman refusing to submit to a man's authority and aggressively asserting her independence. Gremio, a suitor to Bianca, profoundly reveals his opinions of Katherine. He claims "She's too rough for me" (1.1.55). He also professes, "though her/ father be very rich, any man is so very a fool to be/ married to hell" (1.1.122-124). Katherine is undaunted by Gremio's harsh words. Katherine's appearance as a shrew results from underlying insecurities. Katherine feels somewhat inferior to her sister Bianca. Compounded on top of this is Baptist's clear preference for Bianca, He clearly shows favoritism towards Bianca. Katherine says, "Nay, now I see/ She is your treasure, she must have a husband/ I must dance barefoot on her wedding-day, / and for your love to her lead apes in hell. Talk not to me: I will go sit and weep, / Till I can find occasion of revenge" (2.1.31-36). Katherine's relationship with her father is not always on the best of terms.
As Petruchio and Katherine begin their relationship together, Petruchio takes on the difficult chore of taming his shrew. Petruchio's main objective is to bring Katherine into conformity with the acceptable social image of a marriageable young woman.