SET, RELATION and FUNCTION 2021-2018

00:00 Introduction 0:19 - Question 23: What is the period of the function $f(x) = \ln(2 + \sin^2 x)$? 03:10 - Question 37: Analysis of set properties (Distributive law, De Morgan's Law, and conditions for set equality). 07:09 - Question 38: Given sets $X, Y, Z$ with 6, 5, and 4 distinct elements respectively, how many proper subsets does the set formed by the union of $(X - Y)$ and $Z$ have? 12:49 - Question 39: Evaluating statements about relations and functions (specialized relations, Cartesian products). 15:06 - Question 43: Finding the cardinality of the union of $A$ and $B$, where $A$ is the first 250 multiples of 3 and $B$ is the first 200 even natural numbers. 20:03 - Question on Range: What is the range of $f(x) = 1 - \sin x$? 20:35 - Question 35: Comparing sets $A = {1, 3, 5}$ and $B = {2, 4, 7}$ for equivalence and equality. 23:08 - Question 36: Identifying true statements about null sets, subsets of self, and power set cardinality for 10 elements. 24:48 - One-to-one and Onto Check: Analyzing $f(x) = x + 1$ over the domain of integers and natural numbers. 29:56 - Relation Elements: Finding the number of elements in the relation $R$ where $2x + 3y$ equals 20, for natural numbers $x$ and $y$. 32:30 - Question 51: If $f(x + 1) = x^2 - 3x + 2$, what is $f(x)$? 33:46 - Question 95: What is the domain of the exponential function $f(x) = 3^x$? 35:18 - Question 11: Investigating the properties of the relation $xRy$ defined by $\log_a x$ is greater than $\log_a y$ where $a = 0.5$. 41:16 - Question 10: Identifying the correct inequality for $x$ in the range from 1.5 to 4.5. 43:09 - Venn Diagram Analysis (Questions 16-18): 43:41 - Finding the value of $b$ using the ratio of elements in $Y$ and $Z$. 44:41 - Calculating the total elements in the union of $X, Y,$ and $Z$. 45:26 - Determining elements in the complement of $X$. 46:34 - Question 47: How many proper subsets of ${1, 2, 3, 4}$ are supersets of ${3}$? 48:24 - Question 86: If $f(x) = 2x - x^2$, find the value of $f(x+2) + f(x-2)$ at $x=0$. 49:51 - Question 84: What is the domain of inverse cosine of $(x - 2)$? 51:28 - Question 89: What is the minimum value of the absolute value of $(x - 1)$ for real numbers? 54:37 - Question 19: Analyzing the graph and properties of $y = 1 / (x - 1)$. 57:34 - Question 3: Identifying which set relation involving union and intersection is incorrect. 59:03 - Question 7: Solving the inequality where the absolute value of $(x^2 - 3x + 2)$ is greater than $(x^2 - 3x + 2)$. 01:02:38 - Question 9: Finding the minimum number of elements in the union of $A$ and $B$ given the size of each set. 01:04:36 - Question 20: Finding the intersection of set $A$ (where $x$ is between 0 and 2) and set $B$ (where $y$ is a prime number). 01:05:55 - Question 19: Percentage problem involving students playing cricket and football. 01:07:06 - Question 28: Solving the inequality where the product of $(x - a)$ and $(x - b)$ is greater than zero. 01:10:08 - Question 35: Visualizing and solving the intersection of several linear inequalities. 01:11:55 - Question 36: Solving the logarithmic inequality $x$ raised to the power of $\log_7 x$ is greater than 7. 01:15:15 - Question 39: Finding the number of subsets for a set of even numbers up to 20. 01:15:51 - Question 75: Identifying a second-degree polynomial given specific values. 01:17:42 - Composition of Functions (Questions 79-80): Evaluating a triple composition of natural log, tangent, and square functions. Evaluating the triple composition $f(f(f(2)))$ for the square function.