Union, intersection, difference, complement—these operations let us combine, compare, and contrast sets. It’s like doing math with Venn diagrams! 🔁
Union, Intersection, and Difference
Suppose A and B are sets.
A union of A and B is the set:
A \cup B = \{x: x \in A \text{ or } x \in B \}
A intersection of A and B is the set:
A \cap B = \{x: x \in A \text{ and } x \in B \}
A difference of A and B is the set:
A - B = \{x: x \in A \text{ and } x \notin B \}
Concept Check 1
Which of the shaded regions matches the following expression?
(A \cup B) \cap C
The correct answer is the third option. We first take the union of sets A and B, which includes all elements that are in either set. Then, we find the intersection with set C, which includes only those elements that are also in set C.
Complements
Suppose A is a set.
A universal set is a larger set that encompasses other sets.
The complement of A, denoted \bar{A}, is the set \bar{A} = U - A.
P = \{2, 3, 5, 7, ...\} \quad \textbf{(prime numbers)}
\bar{P} = \mathbb{N} - P = \{1, 4, 6, ...\}
Concept Check 2
Find \bar{A}, given:
A = \{1,2,3\}, U = \{0,1,2,3,4,5\}
The correct answer is \{0, 4, 5\}. This is the complement of set A with respect to the universal set U. The complement \bar{A} includes all elements in U that are not in A.
