Math Problem Statement

Find the complement of the following sets. Given the universal set U and sets A and X, calculate their complements.

Solution

The problem you're working on asks to find the complement of the following sets.

Here’s the information from the image:

  • Universal set (U): {10,9,8,,1,0,1,2,,9,10}\{-10, -9, -8, \ldots, -1, 0, 1, 2, \ldots, 9, 10\}, which includes integers from 10-10 to 1010.
  • Set A: {0,1,2,3,4,5}\{0, 1, 2, 3, 4, 5\}
  • Set X: {1,2,3,4,5,7,8,9}\{1, 2, 3, 4, 5, 7, 8, 9\}

To find the complement of these sets:

  1. Complement of A (AcA^c): This is the set of elements in UU but not in AA. So, we take all the elements in UU and subtract the elements in AA. Ac=UA={10,9,8,7,6,5,4,3,2,1,6,7,8,9,10}A^c = U \setminus A = \{-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 6, 7, 8, 9, 10\}

  2. Complement of X (XcX^c): This is the set of elements in UU but not in XX. Xc=UX={10,9,8,7,6,5,4,3,2,1,0,6,10}X^c = U \setminus X = \{-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 6, 10\}

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Related Questions:

  1. What are some properties of the complement in set theory?
  2. How can we use Venn diagrams to visualize complements of sets?
  3. How do union and intersection of complements work mathematically?
  4. What is the complement of a universal set?
  5. How do we apply De Morgan’s laws using complements?

Tip: The complement of a set always relates to the universal set it's defined within. Ensure the universal set is clear before calculating complements.

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Math Problem Analysis

Mathematical Concepts

Set Theory
Complements of Sets
Universal Set

Formulas

Complement of a set A: A^c = U \setminus A
Complement of a set X: X^c = U \setminus X

Theorems

Set Complement Theorem
De Morgan’s Laws

Suitable Grade Level

Grades 9-10