No. As you can learn from Wikipedia, "epsom salts" are magnesium sulfate. The salt used in water softeners is a mix of sodium chloride and potassium chloride, and is dissolved in the water to make an electrolyte to facilitate ion transfer.
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Can you expand upon "make an electrolyte to facilitate ion transfer"? I wouldn't have put it like that. L
As a professional chemist, I would defer to your expertise on this. I tend to look at things from a physics perspective.A water softener (at least, the kind we have in my garage) works, as I understand it, by having a material donate electrons to neutralize the mainly magnesium and calcium ions in "hard" water. The neutral atoms then precipitate out. It's essentially a battery with a single pole. So the picture I have in mind is that the brine you make when you pour those big salt pellets into the water softener's tank is the "electrolyte" of that "battery." It's there to give the electrons a way to get from their source (usual an ion-exchange resin) out to the dissolved ions.
Neither of us are inorganic chemists, but the Na/K "brine" is used to regenerate the ion-exchange resin. Mg++/Ca++(aq) exchange with Na+/K+(resin), up to a point because you eventually have no Na+/K+ left. If you flush the resin with a heavy concentration of Na+/K+ you can drive the exchange the other way as per Le Chatelier's principle L (it was "facilitate" that caught my attention)