I'm working on a political science writing question and need an explanation and answer to help me learn.

1. [From the textbook] We say that a social choice system satisfies the criterion of unanimity if an alternative is the unique social choice whenever every ballot has that alternative ranked highest. Prove that if a social choice procedure satisfies Pareto, then it must also satisfy unanimity.

2. [From the textbook] We saw that sequential pairwise voting does not satisfy Pareto. Does it satisfy unanimity? Explain.

3. [From the textbook] A board of directors for a small corporation consists of three members: Derek with 5 votes as the largest shareholder, Christina with 4 votes as the next largest shareholder, and Isabelle with 3 votes as the smallest shareholder. The quota for passage of any issue is 7 votes. Prove that although each of the three directors has a different number of votes, each has equal power in the sense that no one voter can single-handedly pass a motion and any two voter scan pass any motion.