I have problems in understanding few concepts of elementary set theory. I've choosen a couple of problems (from my problems set) which would help me understand this concepts. To be clear: it's not a homework, I'm just trying to understand elementary set's theory concepts by reading solutions.
Problem 1
(I don't understand this; I mean - not at all)
Let $f: A \to B$, where $A,B$ are non-empty sets, and let $R$ be an equivalence relation in set $B$. We define equivalence relation $S$ in $A$ set by condition:
$aSb \Leftrightarrow f(a)R f(b)$ Determine, which inclusion is always true:
(a) $f([a]_S) \subseteq [f(a)]_R$
(b) $[f(a)]_R \subseteq f([a]_S)$
Notes:
$[a]_S$ is an equivalence class
Problem 2
(I suppose, that (a) is true & (b) is false)
Which statement is true, and which is false (+ proof):
(a) If $f: A \xrightarrow{1-1} B$ and $f(A) \not= B$ then $|A| < |B|$
(b) If $|A| < |B|$ and $C \not= \emptyset$ then $|A \times C| < |B \times C|$
Problem 3
(I don't know, how to think about $\mathbb{Q}^{\mathbb{N}}$ and $\{0,1\}^∗$.)
Which sets have the same cardinality:
$P(\mathbb{Q}), \mathbb{R}^{\mathbb{N}},\mathbb{Z}, \mathbb{Q}^{\mathbb{N}}, \mathbb{R} \times \mathbb{R}, \{ 0,1 \}^*, \{ 0,1 \}^{\mathbb{N}},P(\mathbb{R})$
where $\{ 0,1 \}^*$ means all finite sequences/words that contains $1$ and $0$, for example $000101000100$ or $1010101010101$ etc. $P(A)$ is a Power Set.
Problem 4
(I don't understand this; I mean - not at all)
What are: maximum/minimum/greatest/lowest elements in set:
$\{\{2; 3; 3; 5; 2\}; \{1; 2; 3; 4; 6\}; \{3\}; \{2; 1; 2; 1\}; \{1; 2; 3; 4; 5\}; \{3; 4; 2; 4; 1\}; \{2; 1; 2; 2; 1\}\}$
ordered via subset inclusion
Problem 5
How many equivalence relations there are in $\mathbb{N}$ which also are partial order?
These are simple problems, but I really need to understand, how to solve this kind of problems. I would apreciate Your help.