Consider the following sets:
[tex]$\begin{array}{l}
A =\{x \mid x \text { is alive }\} \
F =\{x \mid x \text { is in France }\} \
M =\{x \mid x \text { is a national monument }\} \
W =\{x \mid x \text { is a woman }\}
\end{array}$[/tex]
Which statements are true given the sets above? Check all that apply.
* Paris [tex]$\in[/tex] F
* The Statue of Liberty [tex]$\in[/tex] A
* The Statue of Liberty [tex]$\in[/tex] M
* Abraham Lincoln belongs to none of these sets.
* The Eiffel Tower is in more than one of these sets.
Answer (2)
Paris ∈ F is true because Paris is located in France.
The Statue of Liberty ∈ A is false because the Statue of Liberty is not alive.
The Statue of Liberty ∈ M is true because the Statue of Liberty is a national monument.
Abraham Lincoln belongs to none of these sets is true because he is not alive, not in France, not a national monument, and not a woman.
The Eiffel Tower is in more than one of these sets is true because it is in France and is a national monument. P a r i s ∈ F , T h e St a t u eo f L ib er t y ∈ M , A b r aham L in co l nb e l o n g s t o n o n eo f t h esese t s , T h e E i ff e lT o w er i s inm ore t han o n eo f t h esese t s
Explanation
Understanding the Sets and Statements We are given four sets: A (alive), F (in France), M (national monument), and W (woman). We need to evaluate five statements to determine which are true.
Evaluating Statement 1 Statement 1: Paris ∈ F . Paris is the capital of France, so Paris is in France. Thus, this statement is true.
Evaluating Statement 2 Statement 2: The Statue of Liberty ∈ A . The Statue of Liberty is a statue, not a living being. Thus, this statement is false.
Evaluating Statement 3 Statement 3: The Statue of Liberty ∈ M . The Statue of Liberty is a national monument in the United States. Thus, this statement is true.
Evaluating Statement 4 Statement 4: Abraham Lincoln belongs to none of these sets. Abraham Lincoln was a man, so he is not in the set W (woman). He is not alive, so he is not in set A . He was not in France, so he is not in set F . He is not a national monument, so he is not in set M . Thus, this statement is true.
Evaluating Statement 5 Statement 5: The Eiffel Tower is in more than one of these sets. The Eiffel Tower is in France, so it belongs to set F . The Eiffel Tower is also a national monument, so it belongs to set M . It is not alive, so it does not belong to set A . It is not a woman, so it does not belong to set W . Since it belongs to both F and M , this statement is true.
Final Answer In summary, the true statements are:
Paris ∈ F
The Statue of Liberty ∈ M
Abraham Lincoln belongs to none of these sets.
The Eiffel Tower is in more than one of these sets.
Examples
Understanding sets and set membership is crucial in many areas, such as database management, where you might define sets of customers based on their location, purchase history, or demographics. For example, a store might define a set of customers who have purchased a specific product in the last month to target them with related offers. Similarly, in biology, sets can represent groups of organisms with shared characteristics, aiding in classification and analysis.
The true statements are: Paris ∈ F, The Statue of Liberty ∈ M, Abraham Lincoln belongs to none of these sets, and The Eiffel Tower is in more than one of these sets.
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