Ch 1 revisits relations and functions at a deeper level. Students classify relations (reflexive, symmetric, transitive, equivalence), determine whether functions are injective/surjective/bijective, and study composition and inverses.
Reflexive: (a,a) ∈ R ∀ a ∈ A. Symmetric: (a,b) ∈ R ⇒ (b,a) ∈ R. Transitive: (a,b),(b,c) ∈ R ⇒ (a,c) ∈ R. Equivalence relation: all three properties hold simultaneously.
One-one (injective): different inputs give different outputs. Onto (surjective): every element of co-domain has a pre-image. Bijective: both one-one and onto. Only bijective functions have inverses.
Composition (gof)(x) = g(f(x)). If f: A→B and g: B→C, then gof: A→C. Inverse: f⁻¹ exists if f is bijective. f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
Download: https://ncert.nic.in/textbook/pdf/lemh101.pdf | Complete book Part I: https://ncert.nic.in/textbook/pdf/lemh1ps.zip
A function has an inverse if and only if it is bijective (both one-one and onto). The inverse reverses the mapping: if f(a) = b, then f⁻¹(b) = a.
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