Sets and Venn Diagrams

The symbol ∩ means "and". It gives the intersection: the elements shared by both sets, shown by the overlap on a Venn diagram. The symbol ∪ means "or". It gives the union: every element in either set or in both, shown by the combined area of the circles.

The complement, written B′, holds everything in the universal set that is not in the set. To find how many elements it has, subtract the number in the set from the universal total. For set B with 10 vanilla lovers out of 30, that gives 30 − 10 = 20.

Add the numbers in every region inside the outer box. Include the overlap where the circles cross. Also add any values outside the circles but still inside the box, counting each element once. In the quiz survey, the regions add up to 28 students.

The universal set, labelled U, is everything being considered in the problem. On a Venn diagram it is the outer rectangle drawn around the circles. In the ice cream survey it represents all 30 students, including the ones who sit outside both circles.

A set is a collection of distinct objects with no repeats and no fixed order. A Venn diagram is a picture that uses overlapping circles to show sets and the relationships between them. The set is the group; the Venn diagram displays it.

A set only records which distinct objects belong to it, not the order they are written in. Because the elements are not ranked and none repeat, a different order still describes exactly the same set. Membership is the only thing that matters.