If S = {letters needed to spell the word 'donkey'} and A = {m, o, n, y} then:
(a) Show S and A in a Venn diagram.
(b) Write the improper subset of S.
(c) Find the number of subsets of A.
Answer (2)
The improper subset of S is {d, o, n, k, e, y}.
The number of subsets of A is calculated using the formula 2 n , where n is the number of elements in the set.
Since A has 4 elements, the number of subsets of A is 2 4 .
Therefore, the number of subsets of A is 16 .
Explanation
Understand the problem and provided data We are given two sets: S, which contains the letters needed to spell the word 'donkey', and A, which is a set of letters {m, o, n, y}. We need to (a) represent these sets in a Venn diagram, (b) identify the improper subset of S, and (c) calculate the number of subsets of A.
Draw the Venn diagram (a) To represent S and A in a Venn diagram, we first identify the elements in each set. S = {d, o, n, k, e, y} and A = {m, o, n, y}. The intersection of S and A contains the elements that are in both sets, which are {o, n, y}. The Venn diagram will have two circles, one for S and one for A, with the overlapping region containing {o, n, y}. The remaining elements of S (d, k, e) will be in the S circle only, and the remaining element of A (m) will be in the A circle only.
Identify the improper subset of S (b) The improper subset of a set is the set itself. Therefore, the improper subset of S is {d, o, n, k, e, y}.
Calculate the number of subsets of A (c) The number of subsets of a set with n elements is 2 n . Set A has 4 elements (m, o, n, y). Therefore, the number of subsets of A is 2 4 = 16 .
State the final answer Therefore, the improper subset of S is {d, o, n, k, e, y}, and the number of subsets of A is 16.
Examples
Venn diagrams are useful in many real-life situations, such as comparing different groups of people or objects. For example, a school can use a Venn diagram to visualize students participating in different sports like basketball, soccer, and volleyball. This helps to understand the overlap and unique participation in each sport. Similarly, understanding subsets is crucial in computer science, especially in data analysis and set theory, where you might need to analyze different combinations of data points.
In a Venn diagram, the letters in set S ({d, o, n, k, e, y}) are displayed with the common elements of set A ({m, o, n, y}) shown in the overlapping area. The improper subset of S is {d, o, n, k, e, y}, and the number of subsets of A is calculated to be 16. Therefore, understanding subsets and Venn diagrams helps visualize the relationships between sets effectively.
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