1 votes 1 votes With reference to the above diagram: $f$ is onto, $g$ is onto $\implies \: g.f$ is onto. if $g.f$ is one-to-one $\implies \: g$ and $f$ both need to be one-to-one. if $g.f$ is onto $\implies \: g$ has to be onto, $f$ need not be onto $f$ is bijective and $g$ is bijective $\implies \: g.f$ is bijective Which of the following statements is TRUE? I,II are correct. I,II,III are correct. I,III,IV are correct. I,IV are correct. Mathematical Logic tbb-mathematics-2 + – Bikram asked May 24, 2017 edited Aug 20, 2019 by Lakshman Bhaiya Bikram 304 views answer comment Share Follow See 1 comment See all 1 1 comment reply shaktisingh commented Jan 8, 2020 reply Follow Share what if fog is written in the statements instead of gof? what will be the changes? does our diagram change? 0 votes 0 votes Please log in or register to add a comment.
Best answer 1 votes 1 votes statement II is false. if g.f is one-to-one => g can be one-to-one but f need not be. Bikram answered May 24, 2017 selected Aug 14, 2019 by Bikram Bikram comment Share Follow See 1 comment See all 1 1 comment reply eikansh commented Jan 30, 2020 reply Follow Share Sir, I think the statement "if g.f is one-to-one => g can be one-to-one but f need not be" is wrong. For example, $ A = \{0\}, B = \{1, 2\}, C = \{0\} \\ f(0) = 1, g(1) = 0, g(2) = 0.$ Then $gof$ is one-one but $g$ is not. https://math.stackexchange.com/questions/22572/injective-and-surjective-functions 0 votes 0 votes Please log in or register to add a comment.