\color{blue}\bf{Problem}: Show that the function given by is a one-to-one function Solution: We know that a function is a one-to-one function iff for all Case 1: Let be even integers Case 2: Let be odd integers Case 3: Let be odd and be even then and i.e. is a false statement, therefore is a […]

Problem: Show that the function f:N→Z given by f(x)={x21−x2x is evenx is odd is a one-to-one function

Solution: We know that a functionf:A→B is a one-to-one function iff f(x1)=f(x2)⇒x1=x2 for all x1,x2∈A

Case 1: Let x1,x2 be even integers

f(x1)=f(x2) ⇒x12=x22 ⇒x1=x2

Case 2: Let x1,x2 be odd integers

f(x1)=f(x2) ⇒1−x12=1−x22 ⇒x1=x2

Case 3: Let x1 be odd and x2 be even

then f(x1)=1−x12≤0 and f(x2)=x22≥1

i.e. f(x1)=f(x2) is a false statement, therefore f(x1)=f(x2)⇒x1=x2 is a true statement because hypotheses is false

Case 4: Let x1 be even and x2 be odd

Same as case 3

Hence f(x1)=f(x2)⇒x1=x2 for all x1,x2∈N

is a true statement, and therefore f is a one-to-one function.