Give examples of two functions f : N → Z and g : Z → Z such that g

1.	Give examples of two functions f : N → Z and g : Z → Z such that g : Z → Z is injective but £ is not injective. (Hint: Consider f(x) = x and g (x) = \|x\|).
Answer» As g (x) = g (-x) = \|x\| for all x ∈ Z , ∴ g is not one-one i.e. y is not injective As f : N → Z and g : Z → Z gof : N → Z let x₁, x₂, ∈ N such that gof (x₁) = gof (x₂) ⇒ g (x₁) = g (x₂) ⇒ \| x₁ \| = \| x₂ \| ⇒ x₁= X₂ (both x₁ ,x₂ >0) Hence g o f is injective.

Give examples of two functions f : N → Z and g : Z → Z such that g : Z → Z is injective but £ is not injective. (Hint: Consider f(x) = x and g (x) = |x|).

Answer»

As g (x) = g (-x) = |x| for all x ∈ Z ,

∴ g is not one-one

i.e. y is not injective

As f : N → Z and g : Z → Z gof : N → Z

let x₁, x₂, ∈ N such that gof (x₁) = gof (x₂)

⇒ g (x₁) = g (x₂)

⇒ | x₁ | = | x₂ |

⇒ x₁= X₂ (both x₁ ,x₂ >0)

Hence g o f is injective.