Abstract Algebra: Theory and Applications

Find an example of two nonempty sets $A$ and $B$ for which $A\times B=B\times A$ is true.

Prove $A\cup \mathrm{\varnothing }=A$ and $A\cap \mathrm{\varnothing }=\mathrm{\varnothing }\text{.}$

Prove $A\cup B=B\cup A$ and $A\cap B=B\cap A\text{.}$

Prove $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)\text{.}$

Prove $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)\text{.}$

Prove $A\subset B$ if and only if $A\cap B=A\text{.}$

Prove $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }\text{.}$

Prove $A\cup B=(A\cap B)\cup (A\setminus B)\cup (B\setminus A)\text{.}$

Prove $(A\cup B)\times C=(A\times C)\cup (B\times C)\text{.}$

Prove $(A\cap B)\setminus B=\mathrm{\varnothing }\text{.}$

Prove $(A\cup B)\setminus B=A\setminus B\text{.}$

Prove $A\setminus (B\cup C)=(A\setminus B)\cap (A\setminus C)\text{.}$

Prove $A\cap (B\setminus C)=(A\cap B)\setminus (A\cap C)\text{.}$

Prove $(A\setminus B)\cup (B\setminus A)=(A\cup B)\setminus (A\cap B)\text{.}$

Let $f:A\to B$ and $g:B\to C$ be invertible mappings; that is, mappings such that $f^{-1}$ and $g^{-1}$ exist. Show that $(g\circ f{)}^{-1}=f^{-1}\circ g^{-1}\text{.}$

Prove the relation defined on ${\mathbb{R}}^2$ by $(x_1,y_1)\sim (x_2,y_2)$ if $x_1^2+y_1^2=x_2^2+y_2^2$ is an equivalence relation.

Define a relation $\sim$ on ${\mathbb{R}}^2$ by stating that $(a,b)\sim (c,d)$ if and only if $a^2+b^2\le c^2+d^2\text{.}$ Show that $\sim$ is reflexive and transitive but not symmetric.

Show that an $m\times n$ matrix gives rise to a well-defined map from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m\text{.}$

Find the error in the following argument by providing a counterexample. “The reflexive property is redundant in the axioms for an equivalence relation. If $x\sim y\text{,}$ then $y\sim x$ by the symmetric property. Using the transitive property, we can deduce that $x\sim x\text{.}$”

Define a relation on ${\mathbb{R}}^2\setminus \{(0,0)\}$ by letting $(x_1,y_1)\sim (x_2,y_2)$ if there exists a nonzero real number $\lambda$ such that $(x_1,y_1)=(\lambda x_2,\lambda y_2)\text{.}$ Prove that $\sim$ defines an equivalence relation on ${\mathbb{R}}^2\setminus (0,0)\text{.}$ What are the corresponding equivalence classes? This equivalence relation defines the projective line, denoted by $\mathbb{P}(\mathbb{R})\text{,}$ which is very important in geometry.