1. Problem: Find the domain of the relation $ R = \{(0, 2), (2, 3), (3, 3), (3, 4)\} $. 2. The domain of a relation is the set of all first elements (inputs) of the ordered pairs. 3. Extract the first elements from each pair: $$\{0, 2, 3, 3\}$$ 4. Remove duplicates to get the domain: $$\{0, 2, 3\}$$ 5. Therefore, the domain of $R$ is $\{0, 2, 3\}$. --- 1. Problem: What is a histogram? 2. A histogram is a graphical representation of data using adjacent rectangles where the height of each rectangle represents the frequency of data in that interval. 3. Therefore, a histogram is a set of adjacent rectangles. --- 1. Problem: What is the most frequent occurring observation in a data set called? 2. The most frequent value in a data set is called the mode. --- 1. Problem: What are the observations that divide a data set into four equal parts called? 2. These observations are called quartiles. --- 1. Problem: Define an onto function. 2. An onto function (surjective) is a function where every element in the codomain has at least one pre-image in the domain. --- 1. Problem: If $A = \mathbb{N}$ (natural numbers) and $B = \mathbb{W}$ (whole numbers), find $B - A$. 2. Since $\mathbb{N} = \{1, 2, 3, ...\}$ and $\mathbb{W} = \{0, 1, 2, 3, ...\}$, 3. $B - A = \{0\}$. --- 1. Problem: Find the arithmetic mean by direct method for the data: 200, 225, 350, 375, 270, 320, 290. 2. Arithmetic mean formula: $$\text{Mean} = \frac{\sum x_i}{n}$$ 3. Calculate sum: $$200 + 225 + 350 + 375 + 270 + 320 + 290 = 2030$$ 4. Number of data points $n = 7$. 5. Calculate mean: $$\frac{2030}{7} = 290$$ --- 1. Problem: Define class mark. 2. Class mark is the midpoint of a class interval, calculated as: $$\text{Class mark} = \frac{\text{Lower limit} + \text{Upper limit}}{2}$$ --- 1. Problem: Name two measures of central tendency. 2. Two common measures are mean and median. --- 1. Problem: If $L = \{a, b, c\}$, $M = \{d, e, f, g\}$, find two binary relations in $L \times M$. 2. $L \times M$ is the set of all ordered pairs $(l, m)$ where $l \in L$ and $m \in M$. 3. Two example binary relations: $$\{(a, d), (b, e)\}$$ $$\{(c, f), (a, g)\}$$ --- 1. Problem: If $X = \{1, 4, 7, 9\}$, $Y = \{2, 4, 5, 9\}$, find $Y \cap X$. 2. Intersection $Y \cap X$ is the set of elements common to both: $$\{4, 9\}$$ --- 1. Problem: Find $a$ and $b$ if $(3 - 2a, b - 1) = (a - 7, 2b + 5)$. 2. Equate components: $$3 - 2a = a - 7$$ $$b - 1 = 2b + 5$$ 3. Solve first equation: $$3 - 2a = a - 7 \Rightarrow 3 + 7 = a + 2a \Rightarrow 10 = 3a \Rightarrow a = \frac{10}{3}$$ 4. Solve second equation: $$b - 1 = 2b + 5 \Rightarrow b - 2b = 5 + 1 \Rightarrow -b = 6 \Rightarrow b = -6$$ --- 1. Problem: Define mode. 2. Mode is the value that appears most frequently in a data set. --- 1. Problem: If $A = \{2, 3, 5, 7\}$, $B = \{3, 5, 8\}$, find $A \cup B$. 2. Union $A \cup B$ is all elements in either set: $$\{2, 3, 5, 7, 8\}$$ --- 1. Problem: State De Morgan’s laws. 2. De Morgan’s laws: $$ (A \cup B)' = A' \cap B' $$ $$ (A \cap B)' = A' \cup B' $$ --- 1. Problem: For data $X = \{12, 5, 8, 4\}$, find the harmonic mean. 2. Harmonic mean formula: $$H = \frac{n}{\sum \frac{1}{x_i}}$$ 3. Calculate sum of reciprocals: $$\frac{1}{12} + \frac{1}{5} + \frac{1}{8} + \frac{1}{4} = 0.0833 + 0.2 + 0.125 + 0.25 = 0.6583$$ 4. Number of data points $n = 4$. 5. Calculate harmonic mean: $$H = \frac{4}{0.6583} \approx 6.08$$ --- 1. Problem: Verify $A - B = A \cap B'$ for $U = \{1,2,3,4,...,10\}$, $A = \{1,3,5,7,8\}$, $B = \{1,4,7,10\}$. 2. $A - B = \{3,5,8\}$ (elements in A not in B). 3. $B' = U - B = \{2,3,5,6,8,9\}$. 4. $A \cap B' = \{3,5,8\}$. 5. Both sets are equal, so verified. --- 1. Problem: Calculate variance for data: 10, 8, 9, 7, 5, 12, 8, 6, 8, 2. 2. Variance formula: $$\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$$ 3. Calculate mean: $$\bar{x} = \frac{10 + 8 + 9 + 7 + 5 + 12 + 8 + 6 + 8 + 2}{10} = \frac{75}{10} = 7.5$$ 4. Calculate squared deviations and sum: $$(10-7.5)^2 = 6.25$$ $$(8-7.5)^2 = 0.25$$ $$(9-7.5)^2 = 2.25$$ $$(7-7.5)^2 = 0.25$$ $$(5-7.5)^2 = 6.25$$ $$(12-7.5)^2 = 20.25$$ $$(8-7.5)^2 = 0.25$$ $$(6-7.5)^2 = 2.25$$ $$(8-7.5)^2 = 0.25$$ $$(2-7.5)^2 = 30.25$$ Sum = 68.5 5. Calculate variance: $$\sigma^2 = \frac{68.5}{10} = 6.85$$ --- 1. Problem: Find standard deviation of salaries: 11500, 12400, 15000, 14500, 14800. 2. Mean: $$\bar{x} = \frac{11500 + 12400 + 15000 + 14500 + 14800}{5} = \frac{68200}{5} = 13640$$ 3. Squared deviations: $$(11500 - 13640)^2 = 299, 600$$ $$(12400 - 13640)^2 = 1, 537, 600$$ $$(15000 - 13640)^2 = 1, 849, 600$$ $$(14500 - 13640)^2 = 7, 344, 100$$ $$(14800 - 13640)^2 = 1, 345, 600$$ Sum = 12,376,500 4. Variance: $$\sigma^2 = \frac{12,376,500}{5} = 2,475,300$$ 5. Standard deviation: $$\sigma = \sqrt{2,475,300} \approx 1573.1$$ --- 1. Problem: Prove $(A - B)' = A' \cup B$ for $U = \{1,2,3,4,...,10\}$, $A = \{1,3,5,7,9\}$, $B = \{1,4,7,10\}$. 2. $A - B = \{3,5,9\}$. 3. $(A - B)' = U - (A - B) = \{1,2,4,6,7,8,10\}$. 4. $A' = U - A = \{2,4,6,8,10\}$. 5. $A' \cup B = \{1,4,7,10\} \cup \{2,4,6,8,10\} = \{1,2,4,6,7,8,10\}$. 6. Both sides equal, so proved.