Composite relations are a fundamental concept in mathematics, especially in the context of relations and functions. Let’s delve into composite relations and then discuss properties of relations:

1. Composite Relations:
   • Given two relations
     

     $R$ and $S$, where $R$ is a relation from set $A$ to set $B$, and $S$ is a relation from set $B$ to set $C$, the composite relation  $R\circ S$ is defined as the set of pairs $(a,c)$ such that there exists an element $b$ in set $B$ where
     $$

     $(a,b)$ is in $R$ and $(b,c)$ is in $S$.

   • Symbolically,
     
     $\mathrm{<br>}\mid \text{there exists }\mathrm{<br>}\text{ such that}\text{ and }\in \mathrm{<br>}\}<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal"></span></span></span></span></span>$

   • 
     $R∘S={(a,c)∣there exists b such that (a,b)∈R and (b,c)∈S}.$

   • The composition of relations is associative, meaning 
     $$(R∘S)∘T=R∘(S∘T)

     R,S,T where the compositions are defined.

2. Properties of Relations:
   • Reflexive Property: A relation$R$ on a set
     $$

     A is reflexive if
     $$

     (a,a) belongs to
     $$

     R for every element 
     $\mathrm{<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;">in </span>}$

     
     $\mathrm{<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">A</span></span></span></span>.</span>}$

   • Symmetric Property: A relation $R$ on a set
     $$

     A is symmetric if for every pair
     $$

     (a,b) in
     $$

     R, the pair
     $$

     (b,a) is also in R.

   • Transitive Property: A relation $R$ on a set
     $$

     A is transitive if for any elements
     $$

     a,b,c in
     $$

     A, if (a,b) and (b,c) belong to R, then (a,c) also belongs to R.

   • Anti-symmetric Property: A relation $R$ on a set A
     $$ is anti-symmetric if for any distinct elements 

     
     $$

     a,b in
     $\mathrm{<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">A\; </span></span></span></span>if </span>}$

     
     $$

     (a,b) belongs to
     $\mathrm{<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">R\; </span></span></span></span>then<span\; class="mopen">(</span><span\; class="mord\; mathnormal">b</span><span\; class="mpunct">,</span><span\; class="mspace"></span><span\; class="mord\; mathnormal">a</span><span\; class="mclose">)\; </span>does\; not\; belong\; to </span>}$

     
     $\mathrm{<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">R</span></span></span></span>.</span>}$

   • Irreflexive Property: A relation 
     $$R on a set A is irreflexive if no element is related to itself, i.e.,
     $$

     (a,a)

     R for any element
     $$

     a in
     $$

     A.

   • Asymmetric Property: A relation$R$ on a set
     $$

     A is asymmetric if for any elements
     $\mathrm{<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">a</span><span\; class="mpunct">,</span><span\; class="mspace"></span><span\; class="mord\; mathnormal">b</span></span></span></span>in,\; if<span\; class="mopen">(</span><span\; class="mord\; mathnormal">a</span><span\; class="mpunct">,</span><span\; class="mspace"></span><span\; class="mord\; mathnormal">b</span><span\; class="mclose">)</span>belongs\; to\; <span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">R</span></span></span></span></span>,\; then\; <span\; class="mopen">(</span><span\; class="mord\; mathnormal">b</span><span\; class="mpunct">,</span><span\; class="mspace"></span><span\; class="mord\; mathnormal">a</span><span\; class="mclose">)</span>does\; not\; belong\; to\; R.</span>}$

   • Total Property: A relation
     

     $R$ on a set $A$ is total if for every pair of distinct elements $a,b$ in $A$, either$(a,b)$ or$(b,a)$ (or both) belongs to $R$.

Understanding these properties is crucial for analyzing and manipulating relations in various mathematical contexts, including graph theory, logic, and relational databases.