Algebra Know-It-ALL

And for the null set

∅ ∪ ∅ = ∅

When two sets are congruent, their union is the same as their intersection. This might seem
trivial right now, but there are situations where it’s not clear that two sets are congruent. In
cases like that, you can compare the union with the intersection as a sort of congruence test.
If the union and intersection turn out identical, then you know the two sets in question are
congruent.

Union with the null set

The union of the null set with any nonempty set gives you that nonempty set. For any
nonempty set X, you can write

X∪∅=X

Remember, any element in the union of two sets only has to belong to one of them.

Union of two disjoint sets

When two nonempty sets are disjoint, they have no elements in common, but their union
always contains some elements. Consider again the sets of even and odd whole numbers, Weven
andWodd. Their union is the set of all the whole numbers. So

Weven∪Wodd= {0, 1, 2, 3, 4, 5, ...}

Union of two overlapping sets

Again, let’s look at the same examples of overlapping sets we checked out when we worked
with intersection. First

L= {2, 3, 4, 5, 6}

M= {6, 7, 8, 9, 10}

The union set here contains nine elements:

L∪M= {2, 3, 4, 5, 6, 7, 8, 9, 10}

The number 6 appears in both sets, but we count it only once in the union. (An element can
only “belong to a set once.”) Now look at these:

P= {21, 23, 25, 27, 29, 31, 33}

Q= {25, 27, 29, 31, 33, 35, 37}

The union set in this case is

P∪Q= {21, 23, 25, ..., 33, 35, 37}

Set Union 31