For Events $A$ and $B$, $A \subseteq B$ is equivalent to $A$ implies $B$? Seeking Clarification.

Question

In my probability class, I was told that, if $A$ and $B$ are events, then $A \subseteq B$ is equivalent to $A$ implies $B$. So in other words, if the event A occurs, then this implies that the event B occurs, right?

However, this claim was not justified, and I'm unsure as to how it is true. It seems to me that, If an event $A$ is a subset of an event $B$, then that does not necessarily imply $B$, since the event $B$ may have other outcomes in it. Indeed, it would seem to me that the converse would be true? In other words, the event $B$ occurring would imply that the event $A$ occurred? This is because, since all of the element in $A$ are also in $B$, then if $B$ occurs, then all of the outcomes that make up $A$ must also have occurred?

I would greatly appreciate it if people could please help me understand this.

Answer 1

Roughly, we are conducting an experiment. Before doing it, we set up a sample space, consisting of all possible outcomes of the experiment. Then, sets whose elements are these outcomes, are called events. Once the experiment has been conducted, exactly one of the outcomes occurs. We say that the event $A$ occurs if the outcome was in $A$. Now, let us look at your question.

If the event $A$ is a subset of the event $B$, then when $A$ occurs, one of the elements in $A$ is the outcome of the experiment. But everything that is in $A$, is also in $B$, so that same outcome is in $B$, and hence $B$ has occurred.

On the other hand,

If $B$ occurs, then some element of $B$ is the outcome of the experiment. But, not every element of $B$ is an element of $A$, so it is possible that the outcome of the experiment may not be an element of $A$. Hence, $B$ may have occurred, but not $A$.

Key point, and flaw in your argument : if an event occurs, it does not mean all the outcomes in it also occur. It means, that the single outcome which did occur is in $A$.

EDIT : If you need clarification on what the word "outcome" means, and how on Earth an experiment has only a single outcome but that leads to many events occurring, do not hesitate to ask. I will illustrate with examples, because it is a confusing point.

Answer 2

Let's work on an example to simplify the discussion. Let $B$ be the event that a single die comes less or equal to 5. Let $A$ the event that the outcome is an odd number. Clearly $A \subseteq B$, and $A$ implies $B$.