deploy: 2cfc699

Author: armanbilge

blog/equivalence-vs-equality.html

@@ -128,9 +128,9 @@ <h2 id="equality" class="section"><a class="anchor-link" href="#equality"><svg x
     <p>Despite the non-uniqueness, there is one equivalence relation that stands out: <em>equality</em>.
     Two objects are considered <em>equal</em> when they are <em>indistinguishable</em> to an observer.
     Formally, equality is required to have the <em>substitution property:</em></p>
-    <span class="bulma-has-text-centered"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo>∈</mo><mi>A</mi><mo separator="true">,</mo><mi mathvariant="normal">∀</mi><mi>f</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mi>B</mi><mo stretchy="false">)</mo><mo>:</mo><mi>a</mi><msub><mo>=</mo><mi>A</mi></msub><mi>b</mi><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mo>=</mo><mi>B</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\forall a,b \in A, \forall f \in (A \to B): a=_A b \implies f(a)=_B f(b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∀</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∀</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">=</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7184em;vertical-align:-0.024em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">=</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span></span></span>
+    <span class="bulma-has-text-centered"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo>∈</mo><mi>A</mi><mo separator="true">,</mo><mi mathvariant="normal">∀</mi><mi>f</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mi>B</mi><mo stretchy="false">)</mo><mo>:</mo><mi>a</mi><msub><mo>=</mo><mi>A</mi></msub><mi>b</mi><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mo>=</mo><mi>B</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\forall a,b \in A, \forall f \in (A \to B): a=_A b \implies f(a)=_B f(b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∀</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∀</span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">=</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7184em;vertical-align:-0.024em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel">=</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.1076em;">f</span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span></span></span>
 
-    <p>(Here, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mo>=</mo><mi>A</mi></msub></mrow><annotation encoding="application/x-tex"> =_A </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5169em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">=</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> denotes equality on <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex"> A </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mo>=</mo><mi>B</mi></msub></mrow><annotation encoding="application/x-tex"> =_B </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5169em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">=</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> denotes equality on <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex"> B </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>.)</p>
+    <p>(Here, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mo>=</mo><mi>A</mi></msub></mrow><annotation encoding="application/x-tex"> =_A </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5169em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">=</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> denotes equality on <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex"> A </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mo>=</mo><mi>B</mi></msub></mrow><annotation encoding="application/x-tex"> =_B </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5169em;vertical-align:-0.15em;"></span><span class="mrel"><span class="mrel">=</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> denotes equality on <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex"> B </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span></span></span></span>.)</p>
     <p>Equality is the finest equivalence: whenever two elements are <em>equal</em>, they are necessarily <em>equivalent</em> with respect to every equivalence.</p>
 
     <h2 id="choices-in-libraries" class="section"><a class="anchor-link" href="#choices-in-libraries"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 576 512"><!--! Font Awesome Free 7.2.0 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2026 Fonticons, Inc. --><path fill="currentColor" d="M419.5 96c-16.6 0-32.7 4.5-46.8 12.7-15.8-16-34.2-29.4-54.5-39.5 28.2-24 64.1-37.2 101.3-37.2 86.4 0 156.5 70 156.5 156.5 0 41.5-16.5 81.3-45.8 110.6l-71.1 71.1c-29.3 29.3-69.1 45.8-110.6 45.8-86.4 0-156.5-70-156.5-156.5 0-1.5 0-3 .1-4.5 .5-17.7 15.2-31.6 32.9-31.1s31.6 15.2 31.1 32.9c0 .9 0 1.8 0 2.6 0 51.1 41.4 92.5 92.5 92.5 24.5 0 48-9.7 65.4-27.1l71.1-71.1c17.3-17.3 27.1-40.9 27.1-65.4 0-51.1-41.4-92.5-92.5-92.5zM275.2 173.3c-1.9-.8-3.8-1.9-5.5-3.1-12.6-6.5-27-10.2-42.1-10.2-24.5 0-48 9.7-65.4 27.1L91.1 258.2c-17.3 17.3-27.1 40.9-27.1 65.4 0 51.1 41.4 92.5 92.5 92.5 16.5 0 32.6-4.4 46.7-12.6 15.8 16 34.2 29.4 54.6 39.5-28.2 23.9-64 37.2-101.3 37.2-86.4 0-156.5-70-156.5-156.5 0-41.5 16.5-81.3 45.8-110.6l71.1-71.1c29.3-29.3 69.1-45.8 110.6-45.8 86.6 0 156.5 70.6 156.5 156.9 0 1.3 0 2.6 0 3.9-.4 17.7-15.1 31.6-32.8 31.2s-31.6-15.1-31.2-32.8c0-.8 0-1.5 0-2.3 0-33.7-18-63.3-44.8-79.6z"/></svg></a>Choices in libraries</h2>