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Relations and Functions
Question
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Show that f : A → B and g : B → C are onto, then g of : A → C is also onto. 

Answer

Here f : A → B and g : B → C are onto functions ∴ g o f is defined from A to C. ∴ g is an onto mapping from B → C ∴ to each z ∈ C, there exists y ∈ B such that g (y) = z Again f is an onto mapping from A to B 
∴ to each y ∈ B, there exists y ∈ A such that f(x) = y ∴ to each z ∈ C, there exists x ∈ A such that z = g (y) = g (f(x)) = (g o f) (x) ∴ g o f is onto.

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Some More Questions From Relations and Functions Chapter

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Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R’ = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

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Give an example of a relation which is

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(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2) : L1 is perpendicular to L2}. Show that R is symmetric but neither reflexive nor transitive.