Venn Diagrams and Operations on Sets

Reading Venn diagrams

A set description tells us which objects belong. A Venn diagram makes relationships between those collections visible. The positions and shaded regions represent membership, so each region must agree with the set definition before we use the picture to reason.

A rectangle usually represents U, the universal set. A circle or another closed curve inside it represents a subset. An object written inside a curve belongs to that set. A member of U outside the curve still belongs to the universe but not to that subset. The physical size of a drawn circle does not by itself tell us the number of elements.

For U = {1, 2, …, 10} and A = {2, 4, 6, 8, 10}, write the even numbers inside A and the odd numbers outside A but inside U. If B = {4, 6}, its curve belongs entirely inside A, expressing B ⊂ A. Keep 4 and 6 in B; they are then automatically also inside A.

When two circles overlap, the overlap represents members common to both sets. If two sets have no common members, draw separate curves. Diagrams can also reason about infinite collections without listing every member: the region then stands for a membership condition.

Union: in either set or both

Sometimes we want everyone who belongs to at least one of two collections. That is what union does. It gathers the members together but still records each distinct object only once, because the result is a set.

Intersection: in both sets

A different question asks which members the sets share. Intersection retains only those passing both membership tests. Being in one set is insufficient; the object must belong to each of the two sets.

The positive even integers and positive odd integers are disjoint, since no integer can be both even and odd. But {1, 2} and {2, 3} are unequal without being disjoint: their intersection is {2}. Always test for a shared member instead of comparing the names of the sets.

Difference: in the first set but not the second

Difference removes from the first set every member that also belongs to the second. The order matters because it determines which collection we start with. It is a membership filter rather than arithmetic subtraction of the listed numbers.

The union separates into three mutually disjoint parts: A − B, A ∩ B, and B − A. A member cannot be in two of those parts at once, because “in B” and “not in B”, or “in A” and “not in A”, cannot both hold. Together the parts account for every member belonging to at least one of A or B.

Properties of union and intersection

These operations satisfy dependable rules because their membership conditions do. Learn what each property means before using its name. That makes it easier to recognise a valid simplification and to avoid importing unrelated arithmetic rules.

Commutative means the order of the inputs can be swapped. Associative means a repeated operation can be regrouped. Idempotent means combining a set with itself returns that set. The empty set is the identity for union, since union with it changes nothing. For intersection, the corresponding unchanged-result role is played by U.

Associativity applies when the same operation is repeated. It does not permit you to drop parentheses from an arbitrary mixture of ∪ and ∩. For a combined expression, compute the parenthesised part first, or use a proved property that justifies rewriting it.

Combining operations and understanding distributivity

A combined expression describes several membership tests. Work from the inside out and explain what each intermediate result represents. Distributivity is useful because it gives a second expression for the same membership requirement.

A member on the left must be in A and also in at least one of B or C. If it is in B, it belongs to A ∩ B; if it is in C, it belongs to A ∩ C. Thus it belongs to the union on the right. Conversely, belonging to either intersection on the right ensures membership in A and in at least one of B or C, so it belongs to the left. We have proved both containments, which proves equality.

To read the corresponding three-circle diagram, first shade B ∪ C, then retain only the shaded part inside A. That gives A ∩ (B ∪ C). Alternatively, shade A ∩ B and A ∩ C separately, then combine those regions. The region common to all three sets remains shaded in both routes. The two constructions identify exactly the same region, not two regions whose areas merely happen to be equal.

At the fourth exercise boundary, finite lists and infinite number sets use the same three definitions. When the expression contains several operations, record an intermediate set before the next operation. This exposes omissions and helps you verify the final membership rule.

Check your understanding

Use the words “either”, “both”, and “first but not second” to interpret the symbol before calculating. For property questions, decide whether the proposed rule follows from that interpretation for all sets.