Find n(A ext ext extbar B), n(B ext ext extbar A), n(A ext ext extbar B^c) and n(A^c ext ext extbar B) given…

Question

Find n(A ext ext extbar B), n(B ext ext extbar A), n(A ext ext extbar B^c) and n(A^c ext ext extbar B) given n(A)=10, n(B)=7, n(A ext ext extbar B)=8 and n(U)=15.

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Here are the solutions to the problems:

1. Find $n(A \cap B)$ if sets $A$ and $B$ have $n(A) = 10$ , $n(B) = 7$ and $n(A \cup B) = 9$ .

Use the formula for the number of elements in the union of two sets.
$n(A \cup B) = n(A) + n(B) - n(A \cap B)$
Substitute the given values into the formula.
$9 = 10 + 7 - n(A \cap B)$
Simplify the equation.
$9 = 17 - n(A \cap B)$
Solve for $n(A \cap B)$ .
$n(A \cap B) = 17 - 9$ $n(A \cap B) = 8$
The number of elements in the intersection of A and B is $\boxed{8}$ .

2. Find the value of $x$ if $\log_x 4 = 2$ .
Convert the logarithmic equation to an exponential equation.
The definition of a logarithm states that $\log_b a = c$ is equivalent to $b^c = a$ .
$x^2 = 4$
Solve for $x$ .
Take the square root of both sides.
$x = \pm\sqrt{4}$ $x = \pm 2$
Consider the domain of the logarithm.
The base of a logarithm must be positive and not equal to 1. Therefore, $x > 0$ and $x \ne 1$ .
$x = 2$
The value of $x$ is $\boxed{2}$ .

3. Simplify $\frac{\sqrt[4]{x^9 y^7}}{\sqrt[4]{xy^{-5}}}$ .
Combine the terms under a single fourth root, using the property $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$ .
$\frac{\sqrt[4]{x^9 y^7}}{\sqrt[4]{xy^{-5}}} = \sqrt[4]{\frac{x^9 y^7}{xy^{-5}}}$
Simplify the expression inside the root using exponent rules $\frac{a^m}{a^n} = a^{m-n}$ .
$\sqrt[4]{x^{9-1} y^{7-(-5)}}$ $\sqrt[4]{x^8 y^{7+5}}$ $\sqrt[4]{x^8 y^{12}}$
Rewrite the radical expression using fractional exponents, $\sqrt[n]{a^m} = a^{m/n}$ .
$(x^8 y^{12})^{1/4}$
Apply the exponent to each term inside the parentheses, using $(ab)^n = a^n b^n$ and $(a^m)^n = a^{mn}$ .
$x^{8 \cdot \frac{1}{4}} y^{12 \cdot \frac{1}{4}}$ $x^2 y^3$
The simplified expression is $\boxed{x^2 y^3}$ .

Find n(A ext ext extbar } B), n(B ext ext extbar } A), n(A ext ext extbar } B^c) and n(A^c ext ext extbar } B) given n(A)=10, n(B)=7, n(A ext ext extbar } B)=8 and n(U)=15.

Step-by-step solution

Use the formula for the number of elements in the union of two sets.

Substitute the given values into the formula.

Simplify the equation.

Solve for $n(A \cap B)$ .

Convert the logarithmic equation to an exponential equation.

Solve for $x$ .

Consider the domain of the logarithm.

Combine the terms under a single fourth root, using the property $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$ .

Simplify the expression inside the root using exponent rules $\frac{a^m}{a^n} = a^{m-n}$ .

Rewrite the radical expression using fractional exponents, $\sqrt[n]{a^m} = a^{m/n}$ .

Apply the exponent to each term inside the parentheses, using $(ab)^n = a^n b^n$ and $(a^m)^n = a^{mn}$ .

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