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$$32 \supset 108 + 110$$ $$98 \supset 97 + 116$$ $$71 \supset 31$$ $$98 < 117 < 38 < (31 = 87) < 32 < 84 < (105 \approx 115) < 49 < 113$$ $$(39 = 65 = 68 = 70 \approx 67) \subset 69 \mapsto 90$$ $$95 \supset 98 - 117 + 47$$ $$63 \supset 38 - 115 + 69$$ $$52 \supset 95 + 63 + 49 + 113$$ $$2 < 80 < 52 < (58 \approx 46) < 92 < 93 \le 94$$ $$47 = 78$$ $$\text{75 and 34, where are they?}$$

Bonus question:

$$37 \approx 55 \land 37 \ne 55, \text{why}?$$

I'm sure that someone will find out the solution quickly. Once you figure it out, it becames very easy.

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Partial Answer

I think this puzzle is related to:

The origin of the name of chemical elements

Partial Examples

$(39 = 65 = 68 = 70 \approx 67) \subset 69 \mapsto 90$

$(\text{Yttrium} = \text{Terbium} = \text{Erbium} = \text{Ytterbium} \approx \text{Holmium}) \subset \text{Thulium} \mapsto \text{Thorium}$

Yttrium, Terbium, Erbium and Ytterbium are named after Ytterby Mine in Sweden.
Holmium is named after Stockholm, the capital of Sweden.
Thulium is named after Thule, a mythical region in Scandinavia.
Thorium is named after Thor, the god in Norse mythology.

$2 < 80 < 52 < (58 \approx 46) < 92 < 93 \le 94$

$\text{Helium} < \text{Mercury} < \text{Tellurium} < (\text{Cerium} \approx \text{Palladium}) < \text{Uranium} < \text{Neptunium} \le \text{Plutonium}$

Helium is named after Helios, the god of the Sun.
Tellurium is named after Tellus, the deity of the Earth.
Cerium is named after Ceres, and Palladium is named after Pallas. Both are asteroids in the asteroid belt, located between Mars and Jupiter.

The remaining examples, FYI:

$$\text{Germanium} \supset \text{Hassium} + \text{Darmstadtium}$$ $$\text{Californium} \supset \text{Berkelium} + \text{Livermorium}$$ $$\text{Lutetium} \supset \text{Gallium}$$ $$\text{Californium} < \text{Tennessine} < \text{Strontium} < (\text{Gallium} = \text{Francium}) < \text{Germanium} < \text{Polonium} < (\text{Dubnium} \approx \text{Moscovium}) < \text{Indium} < \text{Nihonium}$$ $$\text{Americium} \supset \text{Californium} - \text{Tennessine} + \text{Silver}$$ $$\text{Europium} \supset \text{Strontium} - \text{Moscovium} + \text{Thulium}$$ $$\text{Tellurium} \supset \text{Americium} + \text{Europium} + \text{Indium} + \text{Nihonium}$$ $$\text{Silver} = \text{Platinum}$$

Question (Unsolved)

$\text{75 and 34, where are they?}$

$\text{Rhenium and Selenium, where are they?}$

Rhenium is named after the river Rhine.
Selenium is named after Selene, the goddess of the Moon.

I couldn't catch any relation between these two :(

Bonus Question

$37 \approx 55 \land 37 \ne 55, \text{why}?$

$\text{Rubidium} \approx \text{Caesium} \land \text{Rubidium} \ne \text{Caesium}, \text{why}?$

Rubidium is named after Latin word rubidus, meaning deep red.
Caesium is named after Latin word caesius, meaning sky-blue.

Both are named after Latin word meaning a color, but they means different color.

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  • $\begingroup$ You are very close. $\endgroup$ – Victor Stafusa Mar 11 at 6:25
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    $\begingroup$ @VictorStafusa Wow, nice puzzle! Very mysterious at first, but very self-confirming when the solution method is found. $\endgroup$ – Rand al'Thor Mar 11 at 8:47

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