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GO Classes Weekly Quiz 12 | Discrete Mathematics | Group Theory, Functions | Question: 6

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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

  1. $P$ or not $R$
  2. $R$ or not $P$
  3. $R$ or not $Q$
  4. $P$ and not $R$
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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.
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