Describe the union for some sets

Question

Let's describe the union for the following sets using operations with predicates:

1. $A=\{2n~| ~n\in\mathbb{N}\}$ , $B=\{2n+1~|~ n\in\mathbb{N}\}$

2. $A=\{x~|~ x\in\mathbb{R}, x>0\}$, $B=\{x~|~ x\in\mathbb{R}, x^2>0\}$

3. $A=\{4k ~|~ k\in\mathbb{Z}\}$, $B=\{k~|~ 4k+1\in\mathbb{Z}\}$, $C=\{k~|~4k+2\in\mathbb{Z}\}$, $D=\{k~|~ 4k+3\in\mathbb{Z}\}$

Answer 1

1. Write out a few members of $A$ and $B$: $A=\{0,2,4,6,\ldots\}$, and $B=\{1,3,5,7,\ldots\}$. It appears that $A$ is the set of even natural numbers, and $B$ is the set of odd natural numbers; can you see why this is true? That means that $A\cup B=\Bbb N$. That’s how I’d write it, but if you’re required to write it in the form $A\cup B=\{n:\varphi(n)\}$ for some predicate $\varphi$, you can write simply $A\cup B=\{n:n\in\Bbb N\}$.

2. Can you see why $A\subseteq B$, and why that means that $A\cup B=B$? That gives you an easy way to write down a description of $A\cup B$ in the form $\{x:\varphi(x)\}$.

3. I have a suspicion that you’ve misquoted this problem. Are you sure that you have the right definitions of $B,C$, and $D$? Or should $B=\{4k+1:k\in\Bbb Z\}$, with similar changes in $C$ and $D$? If I’m right about this, I suggest that you write out the members of $A,B,C$, and $D$ for $k=0,\pm 1,\pm 2,\pm 3$, and $\pm 4$, say, and see if you can see exactly what numbers must belong to $A\cup B\cup C\cup D$. (If I’m wrong, let me know, and I’ll revise this answer.)