Two lines that are respectively perpendicular to two intersecting lines, always intersect each other. If the mappings f:A→B and g:B→C are both bijective, then the mapping gof:A→C is also bijective
A
True
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B
False
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution
The correct option is A True Given f:A→B is bijective ⇒f is one-one and onto. Also, given g:B→C is bijective ⇒g is one-one and onto. Now, we will check whether gof is bijective
Let x,y∈A such that
(gof)(x)=(gof)(y)
⇒g[f(x)]=g[f(y)]
⇒f(x)=f(y) (Since,g is one-one , so g(x)=g(y)⇒x=y
⇒x=y (∵f is one-one)
Hence, gof is one-one. Now, for surjective, let z∈C be an arbitrary element
Since, g is onto , so for z∈C, there exists an element r∈B such that g(r)=z
Also since, f is one so for every x∈A, there is an element r∈B such that f(x)=r