Question

Show that (a) if \(\phi(0,0, \ldots, 0)=0\) and \(\phi(1,1, \ldots, 1)=1\), then \(\min x_{i} \leqslant \phi(\mathbf{x}) \leqslant \max x_{i}\) (b) \(\phi(\max (\mathbf{x}, \mathbf{y})) \geqslant \max (\phi(\mathbf{x}), \phi(\mathbf{y}))\) (c) \(\phi(\min (\mathbf{x}, \mathbf{y})) \leqslant \min (\phi(\mathbf{x}), \phi(\mathbf{y}))\)

Short answer

In summary, we showed that if φ(0,0,...,0) = 0 and φ(1,1,...,1) = 1, the following properties hold true: (a) min 𝑥𝑖 ≤ φ(x) ≤ max 𝑥𝑖 (b) φ(max(𝑥,𝑦)) ≥ max(φ(𝑥), φ(𝑦)) (c) φ(min (x, y)) ≤ min (φ(x), φ(y))

Step-by-step solution

Evaluate φ at the boundaries

We are given that φ(0,0,...,0) = 0 and φ(1,1,...,1) = 1. These two values will be helpful in proving that min 𝑥𝑖 ≤ φ(x) ≤ max 𝑥𝑖.

Apply the definition of minimum and maximum

The minimum of a set of numbers is the lowest value in the set, and the maximum is the highest value. Since φ(0,0,...,0) = 0 and φ(1,1,...,1) = 1, it follows that 0 is the minimum value of φ(x) and 1 is the maximum value.

Show the inequality

We want to show that min 𝑥𝑖 ≤ φ(x) ≤ max 𝑥𝑖. Since we know that 0 is the minimum value and 1 is the maximum value of φ(x), the inequality min 𝑥𝑖 ≤ 0 ≤ max 𝑥𝑖 holds for any input x. Therefore, the inequality min 𝑥𝑖 ≤ φ(x) ≤ max 𝑥𝑖 is true. (b) Show that φ(max(𝑥,𝑦)) ≥ max(φ(𝑥), φ(𝑦))

Apply the maximum function

The maximum function max(x, y) returns the larger of the two values x and y. Therefore, max(𝑥,𝑦) will never be smaller than either x or y.

Apply φ to the maximum function

We want to show that φ(max(𝑥,𝑦)) is greater than or equal to the maximum of the function φ(𝑥) and φ(𝑦). Since φ(max(𝑥,𝑦)) represents the value of φ at the larger input value, it must be greater than or equal to the value of φ at the smaller input value.

Show the inequality

Since max(𝑥,𝑦)≥x and max(𝑥,𝑦)≥y, it follows that φ(max(𝑥,𝑦)) ≥ φ(x) and φ(max(𝑥,𝑦)) ≥ φ(y). Thus, φ(max(𝑥,𝑦))≥max(φ(𝑥), φ(𝑦)). (c) Show that φ(min (x, y)) ≤ min (φ(x), φ(y))

Apply the minimum function

The minimum function min(x, y) returns the smaller of the two values x and y. Therefore, min(𝑥,𝑦) will never be larger than either x or y.

Apply φ to the minimum function

We want to show that φ(min(𝑥,𝑦)) is less than or equal to the minimum of the function φ(𝑥) and φ(𝑦). Since φ(min(𝑥,𝑦)) represents the value of φ at the smaller input value, it must be less than or equal to the value of φ at the larger input value.

Show the inequality

Since min(𝑥,𝑦)≤x and min(𝑥,𝑦)≤y, it follows that φ(min(𝑥,𝑦)) ≤ φ(x) and φ(min(𝑥,𝑦)) ≤ φ(y). Thus, φ(min (x, y)) ≤ min (φ(x), φ(y)).