Relations and functions

Concepts

• Cartesian product from Modern algebra I 10 exercisesThe Cartesian product of A and B is the set of all ordered pairs whose first coordinate is in A and whose second is in B.
• Binary relation 10 exercisesA set of ordered pairs (a,b) specifying which elements of A are related to which elements of B.
• Equivalence relation 10 exercisesA binary relation R on A that is reflexive, symmetric and transitive. It groups the elements of A that behave alike with respect…
• Equivalence class 10 exercisesGiven an equivalence relation R on A and an element a∈ A, the equivalence class [a] is the set of all x∈ A with x R a.
• Injective function 10 exercisesFunction fcolon A→ B in which distinct elements of A have distinct images in B.
• Surjective function 10 exercisesFunction fcolon A→ B whose image is the whole codomain B.
• Bijective function 10 exercisesFunction fcolon A→ B that is both injective and surjective, so each element of B corresponds to exactly one element of A.
• Image and preimage 10 exercisesGiven fcolon A→ B, the image of X⊆ A is the set of values f(x) with x∈ X, and the preimage of Y⊆ B is the set of a∈ A whose value…
• Composition and inverse 30 exercisesThe composition g∘ f applies f first and then g. The inverse f^-1 exists when f is bijective and undoes its action.