Give an example of two subsets A and B of {1,2,3} such that...

Give an example of two subsets $A$ and $B$ of $\{1,2,3\}$ such that all of the following sets are different: $A \cup B$, $A \cup \bar{B}, \bar{A} \cup B, \bar{A} \cup \bar{B}, A \cap B, A \cap \bar{B}, \bar{A} \cap B, \bar{A} \cap \bar{B}$.

Key Concepts

Set Theory

Set theory is a branch of mathematical logic that studies collections of objects, called sets, and the relationships between them. It forms the foundation for understanding various mathematical operations, such as unions, intersections, and complements, which are vital for problems involving subsets and their interactions.

Subsets

A subset is a set whose elements are all contained within another set, known as the universal set. In many problems, including this one, choosing appropriate subsets is key to constructing examples that demonstrate distinct behaviors of various set operations.

Universal Set

The universal set is the entire set of elements under consideration in a given context. It provides the backdrop against which the complement of a subset is defined, ensuring that every element is either in the subset or its complement.

Union of Sets

The union of two sets is a set that contains all elements that are in either set. The union operation is fundamental in set theory, and understanding it is crucial when considering how combining subsets or their complements can yield different outcomes.

Intersection of Sets

The intersection of two sets is the set of elements that are common to both sets. This operation is used to identify overlaps between subsets or between a subset and the complement of another, playing a critical role in distinguishing between different set combinations.

Complement of a Set

The complement of a subset consists of all the elements in the universal set that are not in the subset. This concept is essential for constructing examples where both the subset and its complement are used in various operations, thereby increasing the number of distinct resulting sets.

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