1. Let [tex]$P = \{p: p \in \text{multiples of } 2, p\ \textless \ 10\}$[/tex] and [tex]$Q = \{q: q \in \text{multiples of } 3, q\ \textless \ 10\}$[/tex] be two sets.
i) List the elements of the sets P and Q.
ii) State whether the sets P and Q are disjoint or overlapping. Give reason.
iii) Show the elements of the sets P and Q in a Venn diagram.
iv) List the elements which belong to the sets P as well as Q.
Answer (2)
List the elements of set P (multiples of 2 less than 10): P = { 2 , 4 , 6 , 8 } .
List the elements of set Q (multiples of 3 less than 10): Q = { 3 , 6 , 9 } .
Determine that sets P and Q are overlapping because they share the element 6.
List the elements that belong to both sets P and Q: { 6 } .
Explanation
Problem Analysis We are given two sets, P and Q . Set P contains multiples of 2 that are less than 10, and set Q contains multiples of 3 that are less than 10. We need to list the elements of each set, determine if the sets are disjoint or overlapping, represent the sets in a Venn diagram, and list the elements that belong to both sets.
Listing Elements of Set P First, let's list the elements of set P , which includes all multiples of 2 less than 10. These are 2, 4, 6, and 8. So, P = { 2 , 4 , 6 , 8 } .
Listing Elements of Set Q Next, let's list the elements of set Q , which includes all multiples of 3 less than 10. These are 3, 6, and 9. So, Q = { 3 , 6 , 9 } .
Determining if Sets are Disjoint or Overlapping Now, let's determine if sets P and Q are disjoint or overlapping. Disjoint sets have no elements in common, while overlapping sets have at least one element in common. Comparing the elements of P and Q , we see that the element 6 is present in both sets. Therefore, sets P and Q are overlapping.
Representing Sets in a Venn Diagram To represent the sets in a Venn diagram, we draw two overlapping circles. One circle represents set P , and the other represents set Q . The overlapping region represents the elements that are common to both sets. In this case, the number 6 goes in the overlapping region. The remaining elements of P (2, 4, and 8) go in the part of the P circle that does not overlap with Q . The remaining elements of Q (3 and 9) go in the part of the Q circle that does not overlap with P .
Listing Common Elements Finally, let's list the elements that belong to both sets P and Q . This is the intersection of the two sets, which we already identified as the element 6.
Final Answer Therefore, the elements that belong to both sets P and Q is { 6 } .
Examples
Understanding sets and their relationships, like disjoint or overlapping sets, is crucial in many real-life scenarios. For instance, consider event planning. If you're organizing a school fair, set P could represent the activities suitable for younger children (e.g., face painting, balloon animals), and set Q could represent activities suitable for older children (e.g., a gaming booth, a sports competition). Identifying the overlapping activities (the intersection of P and Q) helps you plan activities that appeal to all age groups, ensuring everyone has a great time. In this case, a magic show (represented by the number 6) might be enjoyed by both younger and older children, making it a perfect choice for the fair.
Set P consists of {2, 4, 6, 8}, and set Q consists of {3, 6, 9}. The sets are overlapping because they share the element 6. The common elements between sets P and Q is {6}.
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