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Find the cardinal number for the following sets 

A = {106, 107, 108, 109,...,142}

B= {106, 108, 110, . . ., 256}

Set A = { 106, 107, 108, 109,...,142 }
subtract from each element 105, we have 1, 2, 3, ... , 37
n(A) = 142 -105 = 37

Set B = { 106, 108, 110, . . ., 256 }
subtract from each element 104 we have 2, 4, 6, ... , 152
152= 2n

so

n(B) = 152/2 = 76

help trying to figure out what to substract/add

64 students in a classical music lecture class were  polled

with the results that

37 like Wolfgang Amadeus  Mozart,

36 like Ludwig von  Beethoven,

30 like Franz Joseph  Haydn,

14 like Mozart and  Beethoven,

21 like Mozart and  Haydn,

14 like Beethoven and  Haydn, and

8 like all three composers

Find the cardinal number for the following sets

A = {114, 115, 116, 117, ...173}

B = {114, 116, 118, ... 414} 

Set A = { 114, 115, 116, 117, ...173 }
subtract from each element 113 we have 1, 2, 3, ... , 60
n(A) = 173-113 = 60

Set B = { 114, 116, 118, ... 414 }
subtract from each element 112 we have 2, 4, 6, ... , 302
302 = 2n

so

n(B) = 302/2 = 151

Find the cardinal number for the following sets.

A  = {104 , 105, 106 , 107 ,..., 141}

B  = {104 , 106 , 108 , ..., 304 }

Set A = {  104 , 105, 106 , 107 ,..., 141 }
subtract from each element 101 we have 1, 2, 3, ... , 40
n(A) = 141-101 = 40

Set B = {  104 , 106 , 108 , ..., 304  }
subtract from each element 102 we have 2, 4, 6, ... , 202
202 = 2n

so

n(B) = 202/2 = 101

Im trying to understand how to do this anyone knows?

I never did this in high school at all I am lost can anyone solve this? I am looking at my book and still confused. 

whats the cardinal number for 101, 102, 103, 104, . . . 128 and 101, 103, 105, . . . 251

The number of distinct elements in a finite set is called its cardinal number.

It is denoted as n(A) and read as ‘the number of elements of the set’.

For example:

(i) Set A = {2, 4, 5, 9, 15} has 5 elements.

Therefore, the cardinal number of set A = 5. So, it is denoted as n(A) = 5.

Set A = {  101, 102, 103, 104, . . . 128  }

subtract from each element 100 we have 1, 2, 3, ... , 28

n(A) = 128-100 = 28

Set B = {  101, 103, 105, . . . 251  }

subtract from each element 100 we have 1, 3, 5, ... , 151

151 = 2n-1

so \(n = \frac{150}{2} = 75\)

n(B) = 75

factorizing quadratic equation

2x² +13x +15=0

(2x +___)(x+___) = 0

\(\begin{array}{|lrcll|} \hline (2x + a)(x+ b) &=& 0 \\ 2x^2 + 2xb+ax+ab &=& 0 \\ 2x^2 + x\underbrace{(a+2b)}_{=13} + \underbrace{ab}_{=15} &=& 0 \quad |& \quad \text{compare with } 2x^2 +13x +15=0 \\\\ (1) & ab &=& 15 \\ & b &=& \frac{15}{a} \\\\ (2) & a+2b &=& 13 \\ & a + \frac{2\cdot 15}{a} \\ & a + \frac{30}{a} &=& 13 \quad | \quad \cdot a \\ & a^2 + 30 &=& 13a \\ & a^2 -13a + 30 &=& 0 \\\\ & (a-10)(a-3) &=& 0 \\\\ & \mathbf{a_1 = 10} & \text{and} & \mathbf{a_2 = 3} \\\\ & b_1 = \frac{15}{10 } && b_2 = \frac{15}{3} \\ & \mathbf{b_1 = 1.5} && \mathbf{ b_2 = 5} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline (2x + 10)(x+1.5) = 0 \\ \text{ or } \\ (2x + 3)(x+ 5 ) = 0 \\ \hline \end{array}\)

Johnny has $12 less than Sarah.
Sarah has $35 less than Carla.
They have $143.
How much money do each of them have?

Let Johnny = j
Let Sarah = s
Let Carla = c

\(\begin{array}{|lrcll|} \hline (1)& s &=& c - 35 \\\\ (2)& j &=& s-12 \quad | \quad s = c-35 \\ & j &=& (c-35) -12 \\ & j &=& c - 47 \\\\ (3)& j+s+c &=& 143 \quad | \quad j = c - 47 \qquad s = c-35 \\ & (c-47)+(c-35)+c &=& 143 \\ & 3c -47 -35 &=& 143 \\ & 3c &=& 143 + 47 + 35 \\ & 3c &=& 225 \quad | \quad : 3 \\ & \mathbf{c} & \mathbf{=}& \mathbf{75} \\\\ & j &=& c-47 \quad | \quad c = 75 \\ & j &=& 75-47 \\ & \mathbf{j} & \mathbf{=}& \mathbf{28} \\\\ & s &=& c - 35 \quad | \quad c = 75 \\ & s &=& 75 - 35 \\ & \mathbf{s} & \mathbf{=}& \mathbf{40} \\\\ \hline \end{array}\)

\(\begin{array}{lr} \text{Johnny} & $28 \\ \text{Sarah} & $40 \\ \text{Carla} & $75 \\ \end{array}\)

What is %40 of %70

\(\begin{array}{|rcll|} \hline \%40 \text{ of } \%70 \\ &=& 40\ \% \times \ 70\% \\ &=& 40\times \ 70\ \% \% \\ &=& 2800\ \% \% \\ &=& 28\times 100\ \% \% \quad | \quad 100\ \% = 1 \\ &=& 28\times 1 \% \\ &=& 28\ \% \\ \hline \end{array} \)

Two different natural numbers are selected from the set

\( \{1, 2, 3, \ldots, 6\}\).

What is the probability that the greatest common factor of these two numbers is one?

Express your answer as a common fraction.

\(\begin{array}{|r|r|r|r|} \hline n & & \text{greatest common factor } = 1 & \text{greatest common factor } \ne 1 \\ \hline 1. & 5,6 & 1 \\ \hline 2. & 4,6 & & 2 \\ 3. & 4,5 & 1 \\ \hline 4. & 3,6 & & 3 \\ 5. & 3,5 & 1 \\ 6. & 3,4 & 1 \\ \hline 7. & 2,6 & & 2 \\ 8. & 2,5 & 1 \\ 9. & 2,4 & & 2 \\ 10. & 2,3 & 1 \\ \hline 11. & 1,6 & 1 \\ 12. & 1,5 & 1 \\ 13. & 1,4 & 1 \\ 14. & 1,3 & 1 \\ 15. & 1,2 & 1 \\ \hline & sum & 11 \\ \hline \end{array}\)

The probability is \(\mathbf{\tfrac{11}{15}}\)