Question

Create a flow proof for each of the following results. In your flow proof, each step gets its own box or line, and each arrow should be labeled with the appropriate justification.

(a) For any sets A and B, P(A) UP(B) CP(AUB). (Recall P(A) is the power set of A.)
(b) For all rational numbers r and s, r+s is rational. By the way, the above result is known as the Closure Property of Rational Numbers under Addition.
(c) For every real number z, if (x - 3)2 < 0, then 10 - 22 > 0

Answer 1

Flow proofs were constructed for the union of power sets, the closure property of rational numbers under addition, and a result based on the properties of squares for real numbers.

Step-by-step explanation:

To demonstrate the provided mathematical concepts, we create flow proofs for each:

1. For any sets A and B, P(A) ∪ P(B) ⊂ P(A ∪ B). This states that the union of the power sets of A and B is a subset of the power set of their union.
2. For all rational numbers r and s, r+s is rational. This is known as the Closure Property of Rational Numbers under Addition, meaning that the sum of two rational numbers is also a rational number.
3. For every real number z, if the equation (x - 3)2 < 0 is true, then 10 - 22 > 0 must also be true. This is based on properties of squares and the fact that squaring a real number always gives a non-negative result.