Math Problem Statement

A={1,2,{3,4},{5,6,7}}

Select the true statement.

Group of answer choices

{1,2}⊆A

{3}∈A

{3,4}⊆A

{1,2}∈A

Solution

Let's go through the given set A={1,2,{3,4},{5,6,7}}A = \{1, 2, \{3, 4\}, \{5, 6, 7\}\} and evaluate each statement:

  1. {1,2}A\{1, 2\} \subseteq A:

    • This statement is asking whether the set {1,2}\{1, 2\} is a subset of AA. For this to be true, every element of {1,2}\{1, 2\} must be an element of AA.
    • The set AA contains the elements 11, 22, {3,4}\{3, 4\}, and {5,6,7}\{5, 6, 7\}. Since 11 and 22 are elements of AA, this statement is true.
  2. {3}A\{3\} \in A:

    • This statement asks whether the set {3}\{3\} is an element of AA. Notice that AA contains {3,4}\{3, 4\}, but not {3}\{3\}. So, this statement is false.
  3. {3,4}A\{3, 4\} \subseteq A:

    • This statement asks whether the set {3,4}\{3, 4\} is a subset of AA. For this to be true, both 33 and 44 must be elements of AA. However, 33 and 44 are contained in the set {3,4}\{3, 4\}, which is an element of AA, but 33 and 44 are not direct elements of AA. So, this statement is false.
  4. {1,2}A\{1, 2\} \in A:

    • This statement asks whether the set {1,2}\{1, 2\} is an element of AA. Since AA contains the elements 11, 22, {3,4}\{3, 4\}, and {5,6,7}\{5, 6, 7\}, and {1,2}\{1, 2\} is not one of the elements in AA, this statement is false.

Conclusion:

The true statement is:
{1,2}A\{1, 2\} \subseteq A

Would you like further explanation on these concepts? Here are some related questions you might find interesting:

  1. What is the difference between a set being a subset and an element of another set?
  2. How do you distinguish between an element and a subset when working with sets?
  3. What is the meaning of \in and \subseteq in set theory?
  4. How do you determine if a set is a proper subset of another set?
  5. What is the difference between {1,2}\{1, 2\} and {1},{2}\{1\}, \{2\} in set notation?

Tip: Always remember that \subseteq means "is a subset of" (including the possibility of being equal), while \in means "is an element of".

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Math Problem Analysis

Mathematical Concepts

Set Theory
Subset
Element of a Set

Formulas

-

Theorems

Subset Definition
Element of a Set Definition

Suitable Grade Level

Grades 8-10