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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.

  • 0
    I don't understand problem 1 & 4 (I mean - not at all). I suppose, that in 2. (a) is true & (b) is false, but I'm not sure about that. In 3. I don't know, how to think about $\mathbb{Q}^{\mathbb{N}}$ and $\{ 0,1 \}^*$. I think, that in 5. there is a question about equality relation, but - again - I'm not sure.2011-01-12
  • 5
    This question appears to be off-topic because it consists of multiple, unrelated questions.2013-11-28

3 Answers 3