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Chapter 2: Problem 48

Determine whether each statement is true or false. \(6 \in\\{2,4,6,8,10\\}\)

Short Answer

Expert verified
True, 6 is an element of the set \{2,4,6,8,10\}.

Step by step solution

01

Identify the set and the element

The set here is \{2,4,6,8,10\} and the element in question is 6.
02

Check if the element is part of the set

Visually inspect the set or list to verify if the element in question, which is 6, is a part of the set.
03

Formulate the conclusion

After inspecting the set, it's observed that 6 is indeed a part of the set \{2,4,6,8,10\}.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Element Identification
Before you can determine whether a statement is true or false, identify the elements involved.
Focus on two main components: the set and the element in question. In our example exercise, the set is \( \{2,4,6,8,10\} \) and the element we are interested in is 6.
Understanding what a set is can clarify this step. A set is a collection of distinct objects, considered as an object in its own right.
When identifying elements, ensure you are clear about:
  • The composition of the set (What are its members?)
  • The specific element being queried (Which number or object are you examining?)
By properly identifying these components, you set a strong foundation for correctly performing set membership tasks.
Visual Inspection
Once you have identified the element and the set, the next step is visual inspection.
This technique involves looking at the set (in this case, \( \{2,4,6,8,10\} \)) to see if the element (6) is present.During visual inspection,:
  • Carefully scan through each element of the set.
  • Check if you find the element in question among these members.
Visual inspection is often the quickest way to verify set membership, especially with small sets.
It requires attention to detail to ensure no element is overlooked. Practice will improve your skill in quickly conducting accurate inspections.
Conclusion Formulation
After visually inspecting the set, you should now formulate a conclusion.
This involves deciding if the statement concerning the element and set is true or false.In our example, after inspecting the set \( \{2,4,6,8,10\} \), it is evident that the element 6 is indeed a part of the set. Therefore, the statement "\( 6 \in \{2,4,6,8,10\} \)" is true.
When forming conclusions, remember:
  • Base your decision on clear observations from the inspection.
  • Ensure that your conclusion aligns with the results of your inspection.
Being precise in conclusion formulation is critical to ensuring logical and mathematically sound responses.

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Most popular questions from this chapter

Use the formula for the cardinal number of the union of two sets to solve Exercises 93-96. Set \(A\) contains 12 numbers and 18 letters. Set \(B\) contains 14 numbers and 10 letters. One number and 6 letters are common to both sets \(A\) and \(B\). Find the number of elements in set \(A\) or \(\operatorname{set} B\).

In Exercises 41-66, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \(A^{\prime} \cup B^{\prime}\)

In Exercises 41-66, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \(A^{\prime} \cup C^{\prime}\)

Let $$ \begin{aligned} U &=\\{1,2,3,4,5,6,7\\} \\ A &=\\{1,3,5,7\\} \\ B &=\\{1,2,3\\} \\ C &=\\{2,3,4,5,6\\} \end{aligned} $$ Find each of the following sets. \((A \cap B) \cup(A \cap C)\)

A survey of 75 college students was taken to determine where they got the news about what's going on in the world. Of those surveyed, 29 students got the news from newspapers, 43 from television, and 7 from both newspapers and television. Of those surveyed, a. How many got the news from only newspapers? b. How many got the news from only television? c. How many got the news from newspapers or television? d. How many did not get the news from either newspapers or television?
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