Question 1a) If { C $}$ And { D $}$ Are Disjoint, Simplify If Possible { (C \cup D)^{\prime}$}$.b) Express ${ 2 . \overline{143}\$} In The Form Of { \frac{a}{b}$}$ Where { A$}$ And

Mar 9, 2025 by ADMIN 181 views

**Simplifying Complement of Union of Disjoint Sets and Expressing Repeating Decimals as Fractions**

1. Simplifying Complement of Union of Disjoint Sets

1a) Simplifying ${$ (C \cup D)^{\prime}$}$ when ${$ C $}$ and ${$ D $}$ are disjoint

When two sets ${$ C $}$ and ${$ D $}$ are disjoint, it means that they have no elements in common. In other words, their intersection is the empty set, denoted as ${$ \emptyset $}$. The union of two disjoint sets is simply the combination of their elements, without any duplicates.

Given that ${$ C $}$ and ${$ D $}$ are disjoint, we can simplify the expression ${$ (C \cup D)^{\prime}$}$ as follows:

${$ (C \cup D)^{\prime} = C^{\prime} \cap D^{\prime}$]

This is because the complement of the union of two sets is equal to the intersection of their complements.

1b) Understanding the Complement of a Set

The complement of a set [$ A $}$, denoted as ${$ A^{\prime} $}$, is the set of all elements that are not in ${$ A $}$. In other words, it is the set of elements that are in the universal set but not in ${$ A $}$.

For example, if we have a universal set ${$ U $}$ = ${$ {1, 2, 3, 4, 5} $}$ and a set ${$ A $}$ = ${$ {2, 4} $}$, then the complement of ${$ A $}$ is ${$ A^{\prime} $}$ = ${$ {1, 3, 5} $}$.

1c) Simplifying the Expression

Now, let's simplify the expression ${$ (C \cup D)^{\prime}$] using the fact that [$ C $}$ and ${$ D $}$ are disjoint.

Since ${$ C $}$ and ${$ D $}$ are disjoint, we know that ${$ C \cap D = \emptyset $}$. Therefore, we can simplify the expression as follows:

${$ (C \cup D)^{\prime} = C^{\prime} \cap D^{\prime} = (C \cap D)^{\prime} = \emptyset^{\prime} = U $}$

This shows that the complement of the union of two disjoint sets is equal to the universal set.

2. Expressing Repeating Decimals as Fractions

2a) Expressing ${ $2 . \overline{143}\$}$ as a fraction

A repeating decimal is a decimal number that has a block of digits that repeats indefinitely. For example, ${$2 . \overline{143}$] is a repeating decimal where the block [$143$] repeats indefinitely.

To express a repeating decimal as a fraction, we can use the following method:

Let [$ x = 2 . \overline{143} $}$
Multiply both sides of the equation by 1000 to get ${$ 1000x = 2143 . \overline{143} $}$
Subtract the original equation from the new equation to get ${$ 999x = 2141 $}$
Divide both sides of the equation by 999 to get ${$ x = \frac{2141}{999} $}$

This shows that the repeating decimal ${ $2 . \overline{143}\$] can be expressed as the fraction \[$ \frac{2141}{999}$].

2b) Understanding the Concept of Repeating Decimals

Repeating decimals are a type of decimal number that has a block of digits that repeats indefinitely. For example, [$2 . \overline{143}$] is a repeating decimal where the block [$143$] repeats indefinitely.

Repeating decimals can be expressed as fractions using the method described above. This is because the repeating decimal can be represented as a geometric series, which can be summed to get a fraction.

2c) Expressing Repeating Decimals as Fractions

To express a repeating decimal as a fraction, we can use the following method:

Let [$ x = a . \overline{b_1b_2...b_n} $}$
Multiply both sides of the equation by ${ $10^n \$}$ to get ${$ 10^nx = ab_1b_2...b_n . \overline{b_1b_2...b_n} $}$
Subtract the original equation from the new equation to get ${$ 10^nx - x = ab_1b_2...b_n $}$
Divide both sides of the equation by ${ $10^n - 1 \$}$ to get ${$ x = \frac{ab_1b_2...b_n}{10^n - 1} $}$

This shows that any repeating decimal can be expressed as a fraction using this method.

Conclusion

In this article, we have simplified the expression ${$ (C \cup D)^{\prime}$] when [$ C $}$ and ${$ D $}$ are disjoint, and expressed the repeating decimal ${$2 . \overline{143}$] as a fraction. We have also discussed the concept of repeating decimals and how they can be expressed as fractions using a geometric series.

Q1: What is the complement of a set?

A1: The complement of a set [$ A $}$, denoted as ${$ A^{\prime} $}$, is the set of all elements that are not in ${$ A $}$. In other words, it is the set of elements that are in the universal set but not in ${$ A $}$.

Q2: How do you simplify the expression ${$ (C \cup D)^{\prime}$] when [$ C $}$ and ${$ D $}$ are disjoint?

A2: When ${$ C $}$ and ${$ D $}$ are disjoint, we can simplify the expression ${$ (C \cup D)^{\prime}$] as follows:

[ $(C \cup D)^{\prime} = C^{\prime} \cap D^{\prime} = (C \cap D)^{\prime} = \emptyset^{\prime} = U \$}$

This shows that the complement of the union of two disjoint sets is equal to the universal set.

Q3: How do you express a repeating decimal as a fraction?

A3: To express a repeating decimal as a fraction, we can use the following method:

Let ${$ x = a . \overline{b_1b_2...b_n} $}$
Multiply both sides of the equation by ${ $10^n \$}$ to get ${$ 10^nx = ab_1b_2...b_n . \overline{b_1b_2...b_n} $}$
Subtract the original equation from the new equation to get ${$ 10^nx - x = ab_1b_2...b_n $}$
Divide both sides of the equation by ${ $10^n - 1 \$}$ to get ${$ x = \frac{ab_1b_2...b_n}{10^n - 1} $}$

Q4: What is the difference between a repeating decimal and a non-repeating decimal?

A4: A repeating decimal is a decimal number that has a block of digits that repeats indefinitely. For example, ${$2 . \overline{143}$] is a repeating decimal where the block [$143$] repeats indefinitely.

A non-repeating decimal, on the other hand, is a decimal number that does not have any repeating digits. For example, [$3.14159$] is a non-repeating decimal.

Q5: Can you give an example of a repeating decimal that can be expressed as a fraction?

A5: Yes, the repeating decimal [ $2 . \overline{143}\$] can be expressed as the fraction \[$ \frac{2141}{999}$].

Q6: How do you determine if a decimal is repeating or non-repeating?

A6: To determine if a decimal is repeating or non-repeating, you can look for a pattern in the digits. If the digits repeat indefinitely, then the decimal is repeating. If the digits do not repeat, then the decimal is non-repeating.

Q7: Can you give an example of a non-repeating decimal that cannot be expressed as a fraction?

A7: Yes, the non-repeating decimal [ $3.14159\$] cannot be expressed as a fraction. This is because the decimal representation of \[$ \pi $}$ is an irrational number, which cannot be expressed as a finite decimal or fraction.

Q8: What is the significance of expressing repeating decimals as fractions?

A8: Expressing repeating decimals as fractions is significant because it allows us to perform mathematical operations on decimals in a more precise and efficient way. It also helps us to understand the underlying mathematical structure of decimals and how they relate to fractions.

Q9: Can you give an example of a real-world application of expressing repeating decimals as fractions?

A9: Yes, expressing repeating decimals as fractions is used in various real-world applications, such as:

Calculating interest rates and investments
Determining the area and perimeter of shapes
Solving problems in physics and engineering
Analyzing data and statistics

Q10: How do you simplify complex expressions involving sets and decimals?

A10: To simplify complex expressions involving sets and decimals, you can use various mathematical techniques, such as:

Using set theory and algebraic manipulations
Expressing decimals as fractions
Applying mathematical identities and formulas
Using computational tools and software

By mastering these techniques, you can simplify complex expressions and solve problems in a more efficient and effective way.