This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems: 7. Which of the following is a finite set? A. Set of natural numbers B. Set of all real numbers C. Set of days in a week D. Set of all integers A finite set is a set with a countable number of elements. • The set of natural numbers is infinite (\1, 2, 3, ...\). • The set of all real numbers is infinite. • The set of days in a week is finite (\Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday\), containing 7 elements. • The set of all integers is infinite (\..., -2, -1, 0, 1, 2, ...\). The correct option is C. 8. Given the function f(x) = 4x+2, find f(2). Step 1: Substitute x=2 into the function f(x) = 4x+2. f(2) = 4(2) + 2 Step 2: Perform the multiplication. f(2) = 8 + 2 Step 3: Perform the addition. f(2) = 10 The value of f(2) is 10. 9. Consider the function f(x) = 2x. If the domain is \0, 1, 2\, find the range values. Hence, draw the arrow diagram. Step 1: Calculate the range values by substituting each domain value into the function f(x) = 2x. For x=0: f(0) = 2(0) = 0 For x=1: f(1) = 2(1) = 2 For x=2: f(2) = 2(2) = 4 The range values are \0, 2, 4\. Step 2: Draw the arrow diagram. Domain: \0, 1, 2\ Range: \0, 2, 4\ The arrow diagram shows the mapping from each domain element to its corresponding range element: 0 0 1 2 2 4 10. Out of 17 teachers in a school, 10 teach Economics, and 9 teach Maths. The number of teachers who teach both subjects is twice that of those who teach none of the subjects. Use a Venn diagram to find the numbers that teach: (a) Both subjects. (b) None of the subject. (c) Only one subject. Let E be the set of teachers who teach Economics and M be the set of teachers who teach Maths. Total teachers N = 17. Number of teachers teaching Economics, |E| = 10. Number of teachers teaching Maths, |M| = 9. Let x be the number of teachers who teach both subjects, i.e., x = |E M|. Let y be the number of teachers who teach none of the subjects, i.e., y = |E M|'. From the problem statement, the number of teachers who teach both subjects is twice that of those who teach none: x = 2y The number of teachers who teach only Economics is |E| - x = 10 - x. The number of teachers who teach only Maths is |M| - x = 9 - x. The total number of teachers is the sum of those who teach only Economics, only Maths, both, and none: (10 - x) + (9 - x) + x + y = 17 Step 1: Simplify the equation. 19 - x + y = 17 Step 2: Substitute x = 2y into the simplified equation. 19 - (2y) + y = 17 19 - y = 17 Step 3: Solve for y. y = 19 - 17 y = 2 So, the number of teachers who teach none of the subjects is 2. Step 4: Solve for x using x = 2y. x = 2(2) x = 4 So, the number of teachers who teach both subjects is 4. Step 5: Calculate the number of teachers who teach only one subject. Number of teachers who teach only Economics = 10 - x = 10 - 4 = 6. Number of teachers who teach only Maths = 9 - x = 9 - 4 = 5. Number of teachers who teach only one subject = 6 + 5 = 11. The answers are: (a) Both subjects: 4 (b) None of the subject: 2 (c) Only one subject: 11 That's 2 down. 3 left today — send the next one.