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1. Let $x=\{a,b,c\}$ and $y=\{c,d,e\}$.
a.) What is $x \cup y$? What is $x \cap y$?
b.) Let $r_1=\{(1,2),(3,7),(4,9)\}$ and $r_2=\{(1,1),(1,2),(3,11)\}$. What is $r_1 \cup r_2$? What is $r_1 \cap r_2$? Is $r_1$ a function? Is $r_2$ a function? Is $r_1 \cup r_2$ a function? Is $r_1 \cap r_2$ a function?
c.) What is $\{\emptyset, \{\emptyset\}\} \cup \{\emptyset\}$?
2. Prove that $\{\{\emptyset\}\}$ exists in Zermelo set theory. You may use the following theorem:
Theorem A: $(\exists x)(x=\{\emptyset\})$.
3. Is $X=\{\ldots,-4,-3,-2,-1\}$ with the usual order $\leq$ well-ordered?
4. Consider a new theory called "differential algebra theory" which has a one-place predicate $\partial$ and two two-place predicates $+$ and $\cdot$ along with the following axioms:
Axiom 1: $(\forall x)(\forall y)(\partial(x \cdot y)=(\partial x)\cdot y + x \cdot (\partial y))$
Axiom 2: $(\forall x)(\forall y)(\partial(x+y)=\partial(x)+\partial(y))$
a.) Prove
$$\partial((f \cdot g)\cdot h) =((\partial f)\cdot g+f \cdot(\partial g))\cdot h + (f \cdot g)\cdot (\partial h)$$
in differential algebra theory. (hint: when you use US on Axiom 1, let $x=f \cdot g$ and let $y=h$)
b.) Prove
$$\partial(\partial(f \cdot g)) = (\partial (\partial f)\cdot g + (\partial f) \cdot (\partial g)) + ((\partial f) \cdot (\partial g) + f \cdot (\partial(\partial g)))$$
in differential algebra theory.
