3 votes 3 votes Let $f$ be a function from a set $X$ to a set $Y$. Consider the following statements. $P:$ For each $x \in X$, there exists $y \in Y$ such that $f(x)=y$. $Q$ : For each $y \in Y$, there exists $x \in X$ such that $f(x)=y$. $R$ : There exist $x_{1}, x_{2} \in X$ such that $x_{1} \neq x_{2}$ and $f\left(x_{1}\right)=f\left(x_{2}\right)$. The negation of the statement " $f$ is one-to-one and onto $Y$ " is $P$ or not $R$ $R$ or not $P$ $R$ or not $Q$ $P$ and not $R$ Set Theory & Algebra goclasses_wq12 goclasses set-theory&algebra functions 1-mark + – GO Classes asked May 29, 2022 • edited May 29, 2022 by Lakshman Bhaiya GO Classes 364 views answer comment Share Follow See all 0 reply Please log in or register to add a comment.
4 votes 4 votes Since $f$ is a function from $X$ to $Y,$ So, statement $P$ is redundant. Statement $Q$ is the definition of Onto function. Statement $R$ is the definition of “Not one-one” function. GO Classes answered May 29, 2022 GO Classes comment Share Follow See all 0 reply Please log in or register to add a comment.