Algebra Know-It-ALL

Intersection of two nonempty disjoint sets

When two nonempty sets are disjoint, they have no elements in common, so it’s impossible

for anything to belong to them both. The intersection of two disjoint sets is always the null

set. It doesn’t matter how big or small the sets are. Remember the sets of even and odd whole

numbers,Weven and Wodd? They’re both infinite, but

Weven∩Wodd=∅

Intersection of two overlapping sets

When two sets overlap, their intersection contains at least one element. There is no limit to

how many elements the intersection of two sets can have. The only requirement is that every

element in the intersection set must belong to both of the original sets.

Let’s look at the examples of overlapping sets you saw a little while ago, and figure out the

intersection sets. First, examine these

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

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

Here, the intersection set contains one element:

L∩M= {6}

That means the set containing the number 6, not just the number 6 itself. Now look at these:

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

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

The intersection set in this case contains five elements:

P∩Q= {25, 27, 29, 31, 33}

Now check these sets out:

R= {11, 12, 13, 14, 15, 16, 17, 18, 19}

S= {12, 13, 14}

In this situation, S⊂R, so the intersection set is the same as S. We can write that down as follows:

R∩S=S

= {12, 13, 14}

How about the set W 3 − of all positive, negative, or zero whole numbers less than or equal to 3,

and the set W0+ of all the nonnegative whole numbers?

W 3 −= {..., −5,−4,−3,−2,−1, 0, 1, 2, 3}

W0+= {0, 1, 2, 3, 4, 5, ...}

28 The Language of Sets