Using the set builder notation, define the following terms: Disjoint sets, Difference of sets, Null set, Universal set, Complement sets.
Understand the Problem
The question is asking for definitions of several set theory concepts using set builder notation. This involves specifying the terms disjoint sets, difference of sets, null set, universal set, and complement sets in a formal way.
Answer
i) Disjoint sets: \( \{ x \mid x \in A \cap B \} = \emptyset \). ii) Difference: \( \{ x \mid x \in A, x \notin B \} \). iii) Null set: \( \emptyset \). iv) Universal set: \( \{ x \mid x \text{ is an element} \} \). v) Complement: \( \{ x \mid x \in U, x \notin A \} \).
i) Disjoint sets: Two sets A and B are disjoint if ( { x \mid x \in A \cap B } = \emptyset ). ii) Difference of sets: The difference of two sets A and B is ( { x \mid x \in A \text{ and } x \notin B } ). iii) Null set: The null set is ( { x \mid x \neq x } = \emptyset ). iv) Universal set: The universal set U contains all possible elements, ( { x \mid x \text{ is an element} } ). v) Complement sets: The complement of a set A is ( { x \mid x \in U \text{ and } x \notin A } ).
Answer for screen readers
i) Disjoint sets: Two sets A and B are disjoint if ( { x \mid x \in A \cap B } = \emptyset ). ii) Difference of sets: The difference of two sets A and B is ( { x \mid x \in A \text{ and } x \notin B } ). iii) Null set: The null set is ( { x \mid x \neq x } = \emptyset ). iv) Universal set: The universal set U contains all possible elements, ( { x \mid x \text{ is an element} } ). v) Complement sets: The complement of a set A is ( { x \mid x \in U \text{ and } x \notin A } ).
More Information
Set builder notation is a concise way to specify a set by stating the properties that its members must satisfy. Disjoint sets have no elements in common, the null set is empty, and the complement set consists of all elements not in a specified set.
Tips
Ensure correct use of logical symbols (e.g., ( \cap, \cup, \in, \notin )) in set builder notation.
Sources
- Difference of Sets - Definition, Venn Diagram, Complement - Cuemath - cuemath.com
- Set Theory | Definition, Types, Symbols, Examples & Operation on ... - geeksforgeeks.org
- 5.1: Sets and Operations on Sets - Mathematics LibreTexts - math.libretexts.org
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