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One-to-One Relationship: $1$ input has only $1$ output and vice-versa.

$y = x + 1$ is a relation where for every possible value of $x$, there is exactly one value of $y$ and vice-versa.

Many-to-One Relationship: Many inputs could have the same output. However, $1$ input will result in only $1$ output.

$y = x^2$. In this relations, $x = 2$ and $x = -2$, result in $y = 4$. So, many (more than one) values in $x$ could result in one $y$-value.