What is relations and functions? cartesian product of sets, relations, functions, some functions and their graphs (CLASS 11 CBSE MATHS CHPATER 2 NCERT)

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Welcome to Relations and Functions. Let's start with the Cartesian product of sets. The Cartesian product A times B is the set of all ordered pairs where the first element comes from set A and the second element comes from set B. For example, if A contains elements 1 and 2, and B contains elements a and b, then A times B contains four ordered pairs: 1 comma a, 1 comma b, 2 comma a, and 2 comma b. Now let's understand relations. A relation R from set A to set B is simply a subset of the Cartesian product A times B. It tells us which elements of A are connected to which elements of B. In our example, the relation R contains the pairs 1 comma a and 2 comma b. This means 1 is related to a, and 2 is related to b. The domain of relation R is the set of all first elements, which is 1 and 2. The range of R is the set of all second elements, which is a and b. Now let's learn about functions. A function is a special type of relation with strict rules. In a function from set A to set B, every element in A must map to exactly one element in B. This means two conditions: first, every element in A must have an image in B, and second, no element in A can map to more than one element in B. For example, if 1 maps to a and 2 maps to b, this is a valid function. However, if 1 maps to both a and b, then it's not a function because one element maps to multiple elements. Let's explore some common types of functions and their graphs. The identity function f of x equals x produces a straight line passing through the origin with slope 1. The constant function f of x equals c creates a horizontal line. Linear functions f of x equals a x plus b form straight lines with different slopes and y-intercepts. The modulus function f of x equals absolute value of x creates a V-shaped graph that's symmetric about the y-axis. Each function type has its unique graphical representation that helps us understand its behavior. To summarize what we've learned about relations and functions: The Cartesian product creates ordered pairs from two sets. Relations are subsets of these products that connect elements between sets. Functions are special relations where each input maps to exactly one output. Different types of functions like identity, constant, linear, and modulus functions each have their unique graph patterns. These fundamental concepts form the foundation for understanding more advanced mathematical topics.