Conditional Statements (Implications)

TL;DR

Conditional statements, or implications, are "if-then" statements connecting a hypothesis to a conclusion. They are only false when the "if" part is true and the "then" part is false; otherwise, they're true. We use truth tables to evaluate them and understand their relationship to converses, inverses, and contrapositives.

1. The Mental Model

Think of a conditional statement like a promise: "If you clean your room, then you can have dessert." The promise is only broken if you do clean your room, but don't get dessert. In all other scenarios (you don't clean your room, or you clean your room and get dessert), the promise isn't broken.

2. The Core Material

A conditional statement is written as p → q, where 'p' is the hypothesis (the "if" part) and 'q' is the conclusion (the "then" part). We read it as "if p, then q" or "p implies q".

Understanding Truth Values

The truth value of p → q depends on the truth values of p and q. Here's the key:

Notice the only time p → q is false is when p is true and q is false. This is the "broken promise" scenario.

Related Conditional Statements

There are three important statements related to a conditional p → q:

1. Converse: q → p (If q, then p) - Swaps hypothesis and conclusion.
2. Inverse: ¬p → ¬q (If not p, then not q) - Negates both hypothesis and conclusion.
3. Contrapositive: ¬q → ¬p (If not q, then not p) - Swaps and negates both.

It's crucial to know that a conditional statement and its contrapositive always have the same truth value (they are logically equivalent). The converse and inverse are also logically equivalent to each other, but not necessarily to the original conditional.

Here's how these relationships look:

graph TD
    A["Original: p → q (If it rains, then the ground is wet)"] --> B["Converse: q → p (If the ground is wet, then it rains)"]
    A --> C["Inverse: ¬p → ¬q (If it doesn't rain, then the ground isn't wet)"]
    A --> D["Contrapositive: ¬q → ¬p (If the ground isn't wet, then it didn't rain)"]
    B --- E["Logically Equivalent to Inverse"]
    C --- F["Logically Equivalent to Converse"]
    D --- G["Logically Equivalent to Original"]

Necessary and Sufficient Conditions

Sometimes, you'll hear "necessary" and "sufficient" in relation to conditionals:

• p is a sufficient condition for q: This means if p happens, q will happen. (p → q).
• q is a necessary condition for p: This means p cannot happen unless q also happens. (p → q). If q doesn't happen, p can't either (¬q → ¬p). This is just the contrapositive!

3. Worked Example

Let's take the statement: "If a shape is a square, then it is a rectangle."
Let p = "A shape is a square."
Let q = "A shape is a rectangle."

1. Original (p → q): "If a shape is a square, then it is a rectangle." (This is TRUE).

   • If p is True (it's a square) and q is True (it's a rectangle), then p → q is True.
   • Can p be True (it's a square) and q be False (it's not a rectangle)? No, all squares are rectangles. So the "True → False" case never happens here.

2. Converse (q → p): "If a shape is a rectangle, then it is a square." (This is FALSE).

   • Why? You can have a rectangle (q is True) that isn't a square (p is False). For example, a rectangle with sides 2x3. Here, q is True, p is False, so q → p is False.

3. Inverse (¬p → ¬q): "If a shape is not a square, then it is not a rectangle." (This is FALSE).

   • Why? You can have a shape that is not a square (¬p is True, e.g., a 2x3 rectangle) but is a rectangle (¬q is False). Here, ¬p is True, ¬q is False, so ¬p → ¬q is False. Notice how the converse and inverse have the same truth value.

4. Contrapositive (¬q → ¬p): "If a shape is not a rectangle, then it is not a square." (This is TRUE).

   • Why? If a shape isn't a rectangle (¬q is True), then it certainly can't be a square (¬p is True) because all squares are rectangles. Here, ¬q is True, ¬p is True, so ¬q → ¬p is True. Notice how the contrapositive has the same truth value as the original statement.

4. Key Takeaways

• A conditional statement p → q is read "if p, then q" or "p implies q".
• The only time p → q is false is when p is true and q is false.
• The contrapositive (¬q → ¬p) is always logically equivalent to the original conditional (p → q).
• The converse (q → p) and inverse (¬p → ¬q) are logically equivalent to each other.
• Don't confuse the original statement with its converse; they often have different truth values.
• "p is sufficient for q" means p → q.
• "q is necessary for p" also means p → q.

5. Now Try It

Consider the statement: "If a number is divisible by 10, then it is divisible by 5."
1. Identify the hypothesis (p) and conclusion (q).
2. Determine the truth value of the original statement.
3. Write out the converse, inverse, and contrapositive statements.
4. Determine the truth value for each of the related statements.
Success looks like: Clearly stating p and q, correctly assigning truth values for all four statements, and explaining why they are true or false based on the rules.