Why is the empty set a subset of every set?


Quick Answer

The empty set is a subset of every set because no matter which set you compare it with, it always meets the definition of a subset, according to Math is Fun. A subset of another set is defined as a set whose elements are all elements of the other set too. Because the empty set contains no elements, this condition always holds.

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Full Answer

A set is a collection of things, which are called elements. In math, these elements are often numbers or other mathematical objects. The empty set, also called the null set, is a set that contains no elements.

Set A is called a subset of Set B if every element of Set A is also an element of Set B. The empty set meets this requirement for every other set because there is no element of the empty set that is not also an element of that other set. This condition always holds because the empty set contains no elements at all.

Another way to state this is that if x is an element of the empty set, then x is also an element of Set B. Mathematicians refer to statements like this, which generalize about all the elements of the empty set, as vacuous truths.

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