Explain the truth table logic for the implication 'if f is not differentiable, then f is continuous' | Step-by-Step Solution

Understanding Logical Implication with Mathematical Statements

What We're Solving:

We need to understand how truth tables work for the logical implication "If f is differentiable, then f is continuous" (P → Q), where P = "f is differentiable" and Q = "f is continuous."

The Approach:

Truth tables help us understand when logical statements are true or false in ALL possible scenarios. For implications (if-then statements), we need to examine what happens when both the hypothesis (P) and conclusion (Q) can each be either true or false. This is crucial in mathematics because it helps us understand the logical structure of theorems!

Step-by-Step Solution:

Step 1: Set up our variables

• P: "f is differentiable"
• Q: "f is continuous"
• We're analyzing: P → Q ("If f is differentiable, then f is continuous")

Step 2: Create all possible combinations In logic, we need to consider every possible scenario:

• Case 1: P is True, Q is True
• Case 2: P is True, Q is False
• Case 3: P is False, Q is True
• Case 4: P is False, Q is False

Step 3: Analyze each case with mathematical meaning

Case 1: f IS differentiable AND f IS continuous

• This makes P → Q TRUE
• This aligns with the mathematical theorem we know!

Case 2: f IS differentiable BUT f is NOT continuous

• This would make P → Q FALSE
• Mathematically, this is impossible! If a function is differentiable, it must be continuous.

Case 3: f is NOT differentiable BUT f IS continuous

• This makes P → Q TRUE
• Think of f(x) = |x| at x = 0: continuous but not differentiable there!

Case 4: f is NOT differentiable AND f is NOT continuous

• This makes P → Q TRUE
• The implication doesn't claim anything about what happens when P is false.

Step 4: Understanding why Cases 3 & 4 are true When P is false, the implication P → Q is automatically true regardless of Q. Why? Because the implication only makes a promise about what happens when P is true. If P is false, the promise isn't broken!

The Answer:

The truth table for P → Q is:

• T, T → T (differentiable and continuous ✓)
• T, F → F (differentiable but not continuous ✗ - impossible!)
• F, T → T (not differentiable but continuous ✓ - like |x|)
• F, F → T (neither differentiable nor continuous ✓ - promise not broken)

The implication is FALSE only when we have a differentiable function that's not continuous - which violates a fundamental calculus theorem!

Memory Tip:

Think of an implication like a promise: "If it rains, I'll bring an umbrella." The promise is only broken if it rains AND I don't bring an umbrella. If it doesn't rain, my promise stands regardless of whether I have an umbrella or not! Similarly, "If f is differentiable, then f is continuous" is only false if we find a differentiable function that's not continuous (which is mathematically impossible).