Boolean Symbols and Expressions

Consider a different statement:

If it doesn't rain today, then we will go watch the soccer game today.

This statement, which is written as "if $x$ then $y$", can also be equivalently restated as "We will go watch the soccer game today if it doesn't rain today". (In this example, $x = \;($"$\text{it }$$\text{won't }$$\text{rain }$$\text{today}$"$)$ and $y = \;($"$\text{we }$$\text{will }$$\text{go }$$\text{watch }$$\text{the }$$\text{soccer }$$\text{game }$$\text{today}$"$)$.)

We don't have a logic gate for an "if $x$ then $y$" situation, but notice that this statement is true either if $x$ is false (regardless of what $y$ is,) or if $y$ is true (regardless of what $x$ is.) In other words, the only case when this statement is false is when $x$ is true and $y$ is false (that is, when it doesn't rain, but, despite this, we don't go watch the game.)

This description of the behavior of the "if $x$ then $y$" statement tells us that we can express it as: $x' + y$ (not x, or y).

Conclusion: A statement of the form "if $x$ then $y$" (or "$y$ if $x$") can be expressed with the Boolean expression $x' + y$.