Show that the natural numbers and the set of all odd natural numbers have the same cardinality.
We must create a bijective function from the natural numbers to the set of odd natural numbers. Define a function f with the natural numbers as its domain and the set of odd natural numbers as its codomain by f(x) = 2x + 1. This is obviously a bijection so the size of the natural numbers is the size of the set of odd natural numbers.
In microeconomic theory what is the condition needed for a consumer to maximize their utility subject to a budget constraint?
We need the marginal rate of substitution to be equal to the ratio of prices. That is the slope of the indifference curve is equal to the slope of the budget line.
How do we obtain the estimators b1 and b2 in the regression model y = b1 + b2x + u?
By minimizing the sum of squared errors, which is the least squares estimator.