8.1 How to Prove a ∈ A

Proofs Involving Sets

Example

• Let’s investigate elements of $A=\{x:x\in \mathbb{N}\wedge 7|x\}$.
• This set has form $A=x:P(x)$ where $P(x)$ is the open sentence $(x\in N)\wedge (7|x)$.
• Thus $21\in A$ because $P(21)$ is true.
• Similarly, $7,14,28,35$, etc., are all elements of $A$. But $8\notin A$ (for example) because $P(8)$ is false.
• Likewise $-14\notin A$ because $P(-14)$ is false.