# COMPOSITION OF FUNCTION FROM ARROW DIAGRAM

Problem 1 :

The arrow diagrams defines the functions

Find  f∘f

i) f∘f :

Solution :

 Input is a :f∘f (a) = f [f(a)]= f[b]= cSo, (a, c) Input is b :f∘f (b) = f [f(b)]= f[c]= bSo, (b, b) Input is c :f∘f (c) = f [f(c)]= f[b]= cSo, (c, c) Input is d :f∘f (d) = f [f(d)]= f[d]= dSo, (d, d)

Problem 2 :

The arrow diagrams defines the functions

Find  g∘g

i) g∘g :

Solution :

 Input is 1 :g∘g (1) = g [g(1)]= g[4]= 2So, (1, 2) Input is 2 :g∘g (2) = g [g(2)]= g[3]= 1So, (2, 1) Input is 3 :g∘g (3) = g [g(3)]= g[1]= 4So, (3, 4) Input is 4 :g∘g (4) = g [g(4)]= g[2]= 3So, (4, 3)

Input is 5 :

g∘g (5) = g [g(5)]

= g[5]

= 5

So, (5, 5).

The required relation is

g∘g is { (1, 2) (2, 1) (3, 4) (4, 3) (5, 5) }

Problem 3 :

The arrow diagrams defines the functions

Find  f∘g

i) f∘g :

 Input is a :f∘g (a) = f [g(a)]= f[5]= εSo, (a,ε) Input is b :f∘g (b) = f [g(b)]= f[2]= βSo, (b, β) Input is c :f∘g (c) = f [g(c)]= f[4]= δSo, (c, δ) Input is d :f∘g (d) = f [g(d)]= f[4]= δSo, (d, δ)

The required relation is,

{ (a,ε), (b, β), (c, δ), (d, δ) }

Problem 4 :

The arrow diagrams given define the functions

f : {a,b,c,d,e} → {a,b,c,d,e}

and

g : {a,b,c,d,e,f} → {a,b,c,d,e}.

Find the relation, f ∘ g

Solution :

 Input is a :f∘g (a) = f [g(a)]= f[b]= aSo, (a,a) Input is b :f∘g (b) = f [g(b)]= f[e]= aSo, (b, a) Input is c :f∘g (c) = f [g(c)]= f[e]= aSo, (c, a) Input is d :f∘g (d) = f [g(d)]= f[d]= aSo, (d, a) Input is e :f∘g (e) = f [g(e)]= f[a]= eSo, (e, e) Input is f :f∘g (f) = f [g(f)]= f[e]= aSo, (f, a)

So, the required relation is

(a,a) (b, a) (c, a) (b, a) (d, a) (e, e)  (f, a) }

Problem 5 :

The arrow diagrams given define the functions

f : {1, 2, 3, 4} → {α, β, γ, δ, ε, ζ}

and

g : {1, 2, 3, 4, 5, 6} → {1, 2, 3, 4, 5, 6}.

Find the function for g ∘ g

Solution :

Solution :

 Input is 1 :g∘g (1) = g [g(1)]= g[3]= 5So, (1, 5) Input is 2 :g∘g (2) = g [g(2)]= g[5]= 5So, (2, 5) Input is 3 :g∘g (3) = g [g(3)]= g[5]= 5So, (3, 5) Input is 4 :g∘g (4) = g [g(4)]= g[6]= 3So, (4, 3) Input is 5 :g∘g (5) = g [g(5)]= g[5]= 5So, (5, 5) Input is 6 :g∘g (6) = g [g(6)]= g[3]= 5So, (6, 5)

So, the required relation is

(1, 5) (2, 5) (3, 5) (4, 3) (5, 5) (6, 5) }

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