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Chapter 3: Problem 1

Which of the following sets are equal? a) \(\\{1,2,3\\}\) b) \(\\{3,2,1,3\\}\) c) \(\\{3,1,2,3\\}\) d) \(\\{1,2,2,3\\}\)

Short Answer

Expert verified
All the sets a, b, c, and d are equal.

Step by step solution

01

Analyzing set a

Set a contains the elements \(1, 2, 3\). It's important to remember that in set theory, duplicates don't count, so each number here is unique.
02

Comparing set a with b and c

Sets b and c are \(\{3, 2, 1, 3\}\) and \(\{3, 1, 2, 3\}\) respectively. Reordering the elements in set b gives us \(1, 2, 3\), and reordering the elements in set c also gives us \(1, 2, 3\). The number 3 appears twice in both sets, but in set theory, duplicates are discarded, so sets b and c are equal to set a.
03

Comparing set a with d

Set d is \(\{1, 2, 2, 3\}\). Again, reordering the elements gives us \(1, 2, 2, 3\). But, in set theory, duplicates don't count, so we remove one duplicated '2', and set d becomes \(1, 2, 3\), which is equal to set a.

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Key Concepts

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

Equal Sets
In set theory, two sets are considered equal if they contain exactly the same elements, regardless of the order or repetition of those elements. This means that the sets must have identical members with no omissions or additions.

Examining this concept through examples, consider the sets:
  • Set a: \(\{1, 2, 3\}\)
  • Set b: \(\{3, 2, 1, 3\}\)
  • Set c: \(\{3, 1, 2, 3\}\)
  • Set d: \(\{1, 2, 2, 3\}\)
Although the arrangement and repetition of numbers vary, once duplicates are removed and they are reordered, each set results in \(\{1, 2, 3\}\). Thus, in terms of set theory, all these sets are identical and therefore equal to one another.
Duplicate Elements
A characteristic of sets is that duplicate elements have no effect on the set's identity. When you write a set, it does not matter how many times an element is listed; it only matters that the element is present. This is because a set is defined by its unique elements.

In our examples:
  • Set b: \(\{3, 2, 1, 3\}\) has the number 3 listed twice, but for the purposes of defining the set, it's only counted once.
  • Set c: \(\{3, 1, 2, 3\}\) also lists the number 3 twice, but this does not alter the fundamental set \(\{1, 2, 3\}\).
  • Set d: \(\{1, 2, 2, 3\}\) duplicates the number 2, yet when duplicate elements are removed, it becomes \(\{1, 2, 3\}\).
Understanding duplicate elements simplifies these problem types, focusing only upon the individual, unique elements present.
Reordering in Sets
One of the foundational principles of sets is that the order of the elements does not affect the set's identity. This property allows us to rearrange elements as needed without altering the set itself.

For example:
  • Set a: \(\{1, 2, 3\}\) is already in numerical order.
  • Set b: By reordering \(\{3, 2, 1, 3\}\), we arrange it to \(\{1, 2, 3\}\).
  • Set c: Similarly, rearranging \(\{3, 1, 2, 3\}\) also turns it into \(\{1, 2, 3\}\).
  • Set d: Revaluate \(\{1, 2, 2, 3\}\) (removing duplicate) which naturally translates into the same order, \(\{1, 2, 3\}\).
Therefore, with this principle, all the sets end up as the same set, showing how reordering is an essential part of understanding sets in mathematics.

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Most popular questions from this chapter

A set \(A\) has 128 subsets of even cardinality. (a) How many subsets of \(A\) have odd cardinality? (b) What is \(|A| ?\)

a) Among the strictly increasing sequences of integers that start with 1 and end with 7 are: i) 1,7 ii) \(1,3,4,7\) iii) \(1,2,4,5,6,7\) How many such strictly increasing sequences of integers start with 1 and end with \(7 ?\) b) How many strictly increasing sequences of integers start with 3 and end with \(9 ?\) c) How many strictly increasing sequences of integers start with 1 and end with \(37 ?\) How many start with 62 and end with \(98 ?\) d) Generalize the results in parts (a) through (c).

A fair die is rolled three times and the random variable \(X\) records the number of different outcomes that result. For example, if two 5 's and one 4 are rolled, then \(X\) records two different outcomes. Determine (a) the probability distribution for \(X\); (b) \(E(X)\); and (c) \(\operatorname{Var}(X)\).

Ten ping-pong balls labeled 1 to 10 are placed in a box. Two of these balls are then drawn, in succession and without replacement, from the box. a) Find the sample space for this experiment. b) Find the probability that the label on the second ball drawn is smaller than the label on the first. c) Find the probability that the label on one ball is even while the label on the other is odd.

Prove each of the following results without using Venn diagrams or membership tables. (Assume a universe \(U\).) a) If \(A \subseteq B\) and \(C \subseteq D\), then \(A \cap C \subseteq B \cap D\) and \(A \cup C \subseteq B \cup D .\) b) \(A \subseteq B\) if and only if \(A \cap \bar{B}=\emptyset\). c) \(A \subseteq B\) if and only if \(\bar{A} \cup B=?\).
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