Logic

A logical statement $X$ is a statement that is either true or false.

Let $X$ and $Y$ be logical statements. The statement $X$ is necessary for the statement $Y,$ written $X \Leftarrow Y,$ if and only if $Y$ can only be true when $X$ is true. A necessary condition is expressed as “$X$ if $Y$”. Necessity does not guarantee that $Y$ is true when $X$ is true.

Let $X$ and $Y$ be logical statements. The statement $X$ is sufficient for the statement $Y,$ written $X \Rightarrow Y,$ if and only if $Y$ is true whenever $X$ is true. A sufficient condition is expressed as “$X$ only if $Y$”. Sufficiency does not guarantee that $X$ is necessary for $Y.$

Let $X$ and $Y$ be logical statements. The statements $X$ and $Y$ are logically equivalent, written $X \Leftrightarrow Y,$ if and only if $X$ is both necessary and sufficient for $Y.$ An equivalence is expressed as “$X$ if and only if $Y$”.

Let $X$ be a logical statement. The negation of $X$ is the statement “$X$ is false”. The negation of $X$ is true if and only if $X$ is false.

Let $X$ and $Y$ be logical statements. The conjunction of $X$ and $Y$ is the statement “$X$ and $Y$”. The conjunction of $X$ and $Y$ is true if and only if $X$ is true and $Y$ is true.

Let $X$ and $Y$ be logical statements. The disjunction of $X$ and $Y$ is the statement “$X$ or $Y$”. The disjunction of $X$ and $Y$ is true if and only if at least one of $X$ and $Y$ is true.