image links convention, table of contents fix

Also created a separate buttons page

content/buttons.org

@@ -0,0 +1,7 @@
+#+Title: Buttons 
+
+#+attr_html: :width 81px :height 33px
+#+BEGIN_buttons :width 81px :height 33px
+#+HTML: <img src="../pics/CCLogoColorPop1.gif" alt="CC Logo" style="width: 88px; height: 31px;">
+#+HTML: <img src="../pics/brainmade-88x31-dark.png" alt="brainmade.org" style="width: 88px; height: 31px;">
+#+END_buttons

content/countable_union_of_countable_sets_is_countable.org

@@ -2,11 +2,15 @@
 #+created: [2024-12-05 Thu]  
 #+last_modified: [2024-12-06 Fri]  
 #+author: Dibyashanu Pati
-#+OPTIONS: tex:dvipng
+#+OPTIONS: tex:dvisvgm
 #+OPTIONS: \n:t
 #+OPTIONS: toc:2
 
 * Union of Finite Countable Sets
+:PROPERTIES:
+:CUSTOM_ID: union-of-finite-countable-sets 
+:END:
+
 It is straight forward to show that the union of two countable set is countable.
 Let $S_0$ and $S_1$ be two countable infinite sets(in case when either is finite the proof trivial), that is, 
 
@@ -34,6 +38,10 @@ Either using induction or using a similar argument with the naturals modulo $k$
 
 $S = S_0 \cup S_1 \cup S_2 \hdots \cup S_k$.
 * Larger Unions?
+:PROPERTIES:
+:CUSTOM_ID: larger-unions 
+:END:
+
 We know that an arbitrary union of countable sets is not necessarily countable. 
 Consider as a counterexample the union of all singletons $\{r\}, r \in \mathbb{R}$, this is $\mathbb{R}$ itself which we know not to be countable.   
 
@@ -55,11 +63,14 @@ The first time I encountered this it was supposed to be justified was by observi
 Although this is convincing, a thought process that would lead me to come up with this eluded me until now, I explain this thought process below.
 
 * The Argument 
+:PROPERTIES:
+:CUSTOM_ID: the-argument 
+:END:
 In the modulo $k$ argument we have a constant gap between each consecutive element of a sequence, because of this we are limited by the gap size($k$). So the key idea is constructing such a partitioning of the Naturals is to keep increasing the gap size, but since the gap size is finite at any point we can only have elements from a finite number of the countably infinite sub-sequences at any particular, so we are forced to start subsequent sub-sequences at larger and larger points.
 
 The most simple way to increase the gap size in this way is to keep increasing it by one after each gap. 
 
-[[pics/countable_union_of_countable_sets_is_countable.jpg][image 1]]
+[[../pics/countable_union_of_countable_sets_is_countable.jpg][image 1]]
 
 Look at the $0^{th}$ sequence we just get the [[https://en.wikipedia.org/wiki/Triangular_number][Triangular numbers]] and zero. 
 
@@ -169,6 +180,10 @@ Then
 Since $S: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ is both injective and surjective, it is a bijection, we are done.
 
 * An Interesting Application
+:PROPERTIES:
+:CUSTOM_ID: an-interesting-application 
+:END:
+
 *The set of finite subsets of $\mathbb{N}$ is countable.* 
 Let $A_i$ be set of all subsets of $\mathbb{N}$ containing $i$ elements,  that is with cardinality $i$.
 

content/some_evenings_on_this_planet_are_worth_dying_for.org

@@ -4,6 +4,6 @@
 #+author: Dibyashanu Pati
 #+OPTIONS: tex:dvipng
 
-[[pics/some_evenings_on_this_planet_are_worth_dying_for.jpg][Evening 26 January 2025]]
+[[../pics/some_evenings_on_this_planet_are_worth_dying_for.jpg][Evening 26 January 2025]]
 
 *Some Evenings on this Planet are worth Dying for* - no matter how bleak things are otherwise, something shifts inside my brain when I see Venus peeking through the golden skies like this..

content/we_do_not_surrender_we_resist.org

@@ -14,4 +14,4 @@ What should we do if this happens?
 /In such cases we do not surrender, *we resist*, we resist by *learning* !!/
 #+end_quote
 
-[[pics/we_do_not_surrender_we_resist.jpg][resist]]
+[[../pics/we_do_not_surrender_we_resist.jpg][resist]]