Relations and Functions Class 12 Test Series

What Will You Learn?

• Formal Definition of a Relation:
• You will learn that a relation is a subset of the Cartesian product of two sets, which provides a mathematical way to describe a connection between elements of those sets.
• Analyzing Properties of Relations: You will learn to classify relations based on their specific properties. The key types you'll focus on are:
• Reflexive: A relation where every element is related to itself (e.g., "is equal to").
• Symmetric: A relation where if element 'a' is related to 'b', then 'b' is also related to 'a' (e.g., "is a sibling of").
• Transitive: A relation where if 'a' is related to 'b' and 'b' is related to 'c', then 'a' is related to 'c' (e.g., "is less than").
• Equivalence Relations: You will master the concept of an equivalence relation, which is a relation that is simultaneously reflexive, symmetric, and transitive.
• This is a crucial concept as it helps to partition a set into disjoint subsets called equivalence classes.
• Function as a Special Relation:
• You will understand that a function is a specific, well-behaved type of relation where every input from the domain has exactly one unique output in the codomain.
• Analyzing Types of Functions: You will learn to classify functions based on their mapping properties, which is essential for understanding their behavior and invertibility.
• One-to-one (Injective) Functions: You'll learn to identify functions where every distinct element in the domain maps to a distinct element in the codomain. In other words, no two inputs share the same output.
• Onto (Surjective) Functions: You will understand functions where every element in the codomain is the image of at least one element from the domain. This means the range of the function is equal to its codomain.
• Bijective Functions: By combining the two concepts above, you will learn to identify bijective functions, which are both one-to-one and onto. This is a critical skill because only bijective functions are invertible.
• In essence, this syllabus will equip you with the foundational language and analytical tools to:
• Classify and analyze different types of mathematical relationships.
• Distinguish between a general relation and a more structured function.
• Evaluate the specific behavior of functions, which is a prerequisite for studying concepts like inverse functions, continuity, and differentiability in calculus.