Imagine you have a statement that says, “If it’s raining (p), then I will take an umbrella (q).” This statement is an example of implication.
- ”p” stands for the condition: In this case, it’s raining.
- ”q” stands for the action: The action is taking an umbrella.
So, the implication, written as p⇒q, is saying, “if it’s raining (p), then I will take an umbrella (q).”
When is the Implication True or False?
- The implication is true when the condition (p) is not true or when the action (q) does happen. For example, if it’s not raining, the implication is true regardless of whether you take an umbrella or not.
- The implication is false only when the condition (p) is true, but the action (q) doesn’t happen. In our example, if it’s raining, but you don’t take an umbrella, then the implication is false.
So, the essence is, “if the condition is not true or the action happens, the implication is true; otherwise, it’s false.”
Putting it Together - Implication in Practice:
Let’s consider a real-life scenario:
- Statement: “If it’s your birthday (p), then you will get a present (q).”
- If it’s not your birthday (not p), it doesn’t matter if you get a present or not; the implication is true.
- If it is your birthday (p), and you do get a present (q), the implication is true.
- If it is your birthday (p), but you don’t get a present (not q), the implication is false.
In a Nutshell:
Implication is like saying, “If something happens (p), then something else will happen (q).” It’s true unless the first thing happens, but the second one doesn’t. In all other cases, it holds true.
So, “If it’s clear skies (not raining), then I won’t take an umbrella” - this implication is true. But if it’s raining (p), and you don’t take an umbrella (not q), then the implication is false.