Union, intersection and complement

Japanese school year: Math I

What you learn

You will learn set notation, how to find unions and intersections of sets, and how to determine the complement within a universal set. These fundamental tools are essential across mathematics for classifying objects based on given conditions. Basic knowledge of numbers and inequalities will help you grasp these concepts quickly.

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Key points

This shows that the region outside " $A$ or $B$ " is identical to the overlap between "outside $A$ " and "outside $B$ " (De Morgan's law). Use it to simplify complex set expressions.

\overline{A \cup B} = \bar{A} \cap \bar{B}

Use this formula to expand parentheses and simplify set operations (distributive law). Just like algebraic multiplication, intersecting with $A$ is distributed to both $B$ and $C$ .

A \cap (B \cup C) = (A \cap B) \cup (A \cap C)

This formula states that combining set $A$ and its complement $\bar{A}$ gives the whole universe $U$ , with no overlap between them (empty set $\emptyset$ ). Use it to analyze basic set relationships.

A \cup \bar{A} = U, \quad A \cap \bar{A} = \emptyset

Choose a set to practice.

Practice 1Coming soon