Higher Mathematics · Classes 9-10 · Chapter 1

Difference, Complement, Union and Intersection of Sets.

Give me two sets and I can make new ones from them: take one away from the other, pour them together, or keep only what they share. There are four operations, and each one is a region of a Venn diagram. Once you can see the region, half the question is already answered.

1. The difference of two sets

For sets A and B, A \ B = {x : x ∈ A and x ∉ B}, read “A delete B”. Start with A, throw out every element that also sits in B, and what is left is A \ B.

Everything in A \ B came out of A, and nothing came in from outside. So always A \ B ⊆ A.

Example 4: with A = {0, 1, 2, ···, 9} and B = {0, 2, 4, 6, 8, 10}, the even numbers leave A and A \ B = {1, 3, 5, 7, 9}. The 10 in B changes nothing, because 10 was never in A.

And when the two sets share nothing, there is nothing to throw out. With A = {1, 2, 3} and B = {a, b, c}, A \ B = A.

2. The complement

If U is the universal set and A ⊆ U, then U \ A is the complement of A: U \ A = {x : x ∈ U and x ∉ A}, written A′ or Aᶜ. A complement is just a difference taken from the universal set.

Example 5: if U is the set of all integers and A the set of all negative integers, then A′ = {0, 1, 2, 3, ···}. Notice that 0 is not negative, so 0 belongs to the complement.

A and A′ together fill U, yet they share not a single element: A ∪ A′ = U and A ∩ A′ = ∅. Take the complement of the complement and you are back where you began: (A′)′ = A.

3. Venn diagrams, union and intersection

A picture that shows set information this way is a Venn diagram, named after its inventor John Venn (1834-1923). In the book's figure the rectangle is the universal set U, the circle inside it is A, and the part inside the rectangle but outside the circle is A′.

The union of A and B is A ∪ B = {x : x ∈ A or x ∈ B}: every element of either set. An element that sits in both is still written only once.

The intersection of A and B is A ∩ B = {x : x ∈ A and x ∈ B}: only the elements the two sets have in common.

Example 8: U = {0, 1, ···, 9}, A = {x : x is prime} = {2, 3, 5, 7} and B = {x : x is odd} = {1, 3, 5, 7, 9}. Then A ∪ B = {1, 2, 3, 5, 7, 9}, A ∩ B = {3, 5, 7}, A′ = {0, 1, 4, 6, 8, 9} and B′ = {0, 2, 4, 6, 8}.

In the same example A′ ∪ B′ = {0, 1, 2, 4, 6, 8, 9}, which is exactly (A ∩ B)′, and A′ ∩ B′ = {0, 4, 6, 8}, which is exactly (A ∪ B)′. The next note names this: De Morgan's laws.

The ten numbers of Example 8 sit in their own regions of the diagram. Pick an operation and watch which regions shade and which numbers make the list.

Check every number against every operation. Tap a number to see which sets it lands in, and why.

Two circles cut the rectangle into four regions. Only A is A \ B, also written A ∩ B′. The shared part is A ∩ B. Only B is B \ A. Outside both circles but inside the rectangle is A′ ∩ B′, which is (A ∪ B)′. Every expression is built from these four pieces. Read the expression below, tap the right pieces, then check.

4. Disjoint sets

When A ∩ B = ∅, so the sets have no element in common, A and B are disjoint. Example 9: the positive integers and the negative integers are disjoint. In a Venn diagram the two circles do not touch at all.

For instance, with A = {1, 2, 3} and B = {4, 5, 6}, A ∩ B = ∅ and A ∪ B = {1, 2, 3, 4, 5, 6} = {x : x ∈ N, x < 7}.

Example 10: A = {x ∈ R : 0 ≤ x ≤ 2} and B = {x ∈ N : 0 ≤ x ≤ 2}. B holds only natural numbers, and 0 is not one, so B = {1, 2}. Every element of B lies in A, so B ⊆ A, A ∪ B = A and A ∩ B = B = {1, 2}.

Example 11: A = {x ∈ R : 1 ≤ x ≤ 2} and B = {x ∈ R : 0 < x < 1}. B stops just short of 1, and A starts at 1. Joined, A ∪ B = {x ∈ R : 0 < x ≤ 2}, and yet A ∩ B = ∅: A and B are disjoint.

5. Sets of multiples and sets of intervals

Aₙ = {n, 2n, 3n, ···} is the set of multiples of n. Aₘ ∩ Aₙ holds the numbers that are multiples of both m and n, which are the multiples of their LCM. So A₁ ∩ A₂ = A₂, A₃ ∩ A₆ = A₆ and A₂ ∩ A₃ = A₆.

For sets given by real intervals, draw both on one number line and see where they overlap. With A = {x : 2 < x ≤ 5} and B = {x : 1 ≤ x < 3}, both hold from just after 2 to just before 3, so A ∩ B = {x ∈ R : 2 < x < 3}. Taking U = R, A′ = {x : x ≤ 2 or x > 5}: a closed end of A becomes an open end of the complement, and an open end becomes closed.

Now overlap them yourself. Two intervals share one number line; pick an operation below and its result is drawn as the blue bar. Watch whether each end is filled or hollow.

6. Check yourself

The first comes straight from the definition of a complement.

For the second, list the two sets first.

In the last one, first see whether anything is shared.

The whole note on eight cards.

Source: Higher Mathematics, Classes 9-10 (NCTB), Chapter 1 “Set and Function”, the “Difference, Complement, Union and Intersection of Sets” part.