Venn diagrams are useful visual tools for showing set relationships.
Consist of a rectangle for the universal set and circles for subsets.
Named after John Venn, an English mathematician.
We'll focus on Venn diagrams with two circles; later, we'll look at three circles.
Drawing Venn Diagrams
Basic Structure: Universal set in a rectangle; subsets in overlapping circles.
Disjoint Sets: Circles can be separated if there are no common elements.
Subsets: Overlapping circles if sets share common elements, or nested circles if one is a subset of the other.
Equal Sets: Circles can overlap completely or be represented by a single circle.
Set Operations
Complement
Definition: Elements in the universal set but not in the subset.
Example: If universal set includes A, B, C, D, E, F and subset A includes B, D, F, the complement of A includes A, C, E.
Intersection
Definition: Elements common to both sets, represented by the overlap in a Venn diagram.
Example: Sets G {P, Q, S, T} and H {R, S, T, V} have an intersection of {S, T}.
Union
Definition: All elements in both sets, without duplication of common elements.
Example: Sets V {, @, #} and W {#, ^} have a union of {, @, #, ^}.
Difference
Definition: Elements in the first set that are not in the second set.
Example: Difference of A {A, B, C, D, E} and B {D, E, F, G, H} is {A, B, C}.
Cartesian Product
Definition: Set of ordered pairs derived from two sets.
Example: A {apple, pear} and B {red, blue, yellow} result in {(apple, red), (apple, blue), (apple, yellow), (pear, red), (pear, blue), (pear, yellow)}.
Using Venn Diagrams and Operations
When performing operations: handle parentheses first, then complements, and finally other operations.
Examples:
A Union B Complement: First find A union B, then identify missing elements from the universal set.
A Complement Union B Complement: Find individual complements first, then union them.
Differences: A complement minus B complement yields unique elements not shared in B.
Conclusion
Practice with two-circle operations lays groundwork for three-circle Venn diagrams.
Future topics include verifying equal sets and applying these concepts to real-life word problems.