A water softener does not directly reduce Total Dissolved Solids (TDS) in water. TDS is a measure of the amount of dissolved minerals, salts, and other inorganic substances present in water.
A traditional water softener works by removing the hard water minerals such as calcium and magnesium which are responsible for making the water "hard". It does this by using an ion exchange process where the hard water minerals are exchanged with sodium ions. However, this process doesn't lower the TDS of the water. In fact, it can increase the TDS of the water as the sodium ions present in the resin of the water softener can increase the overall TDS level of the water.
Salt-free water softeners, also known as "water conditioners" or "TAC" (template-assisted crystallization) do not remove the minerals from the water, but rather change the physical properties of the minerals so that they can no longer cause hard water problems. This means that the TDS level of the water is not affected by this type of water softener.
In short, a water softener does not reduce TDS, but it does remove the hard water minerals responsible for making the water hard. The TDS level of the water can be increased by traditional water softeners, but it is not affected by salt-free water softeners.