Exercises for Week 1

1. In the context of naive set theory, using any mathematical objects you wish (for example $\mathbb{N}$, $\mathbb{Q}$, $0$, $1$, $\pi$,)
   1. Give an example of mathematical objects $x$ and $y$ with $x \in y$ but $x \not\subseteq y$.
   2. Give an example of mathematical objects $x$ and $y$ with $x \subseteq y$ but $x \notin y$.
2. In the context of axiomatic set theory, prove that there exist
   1. $x$ and $y$ with $x \in y$ but $x \not\subseteq y$.
   2. $x$ and $y$ with $x \subseteq y$ but $x \notin y$.
   Hint: build $x$ and $y$ from $\emptyset$ using the pairing operation $\{\cdot,\cdot\}$. Then your proof will use the definition of the pairing operation.
3. Recall that the Kuratowski ordered pair $(x,y)$ is defined in terms of the pairing operation as $\{\{x\},\{x,y\}\}$. Prove that $(x_1,y_1) = (x_2,y_2)$ if and only if $x_1 = x_2$ and $y_1 = y_2$. Hint: the axiom of extensionality gives a useful criterion for two sets to be equal.
4. Given $z_1$, $z_2$, and $z_3$, prove that there is a set whose members are $z_1$, $z_2$, $z_3$, and nothing else. (The notation $\{z_1,z_2,z_3\}$ presupposes the existence of the set in question, so you should not use it in your answer.)