如何证明条件逻辑表达式(A&&C)||(B&&C)与(A||B)&&C等价？

Hey there! This is a core question in boolean logic, so let's start with the general approaches to proving conditional expression equivalence, then dive into your specific example.

When verifying if two boolean expressions are equivalent, these are the most reliable go-to methods:

1. Truth Table Verification #

This is the most straightforward approach, especially for expressions with a small number of variables. You list every possible combination of truth values (true/false) for all variables, compute the result of each expression for each combination, and check if the results match across all rows. If they do, the expressions are equivalent.

2. Boolean Algebra Transformation #

Use well-established boolean laws (distributive, commutative, associative, etc.) to rewrite one expression step-by-step until it matches the other. This is more efficient for complex expressions once you’re comfortable with the rules.

3. Logical Intuition & Deduction #

Break down what each expression means in plain language, then reason through whether they describe the exact same condition. For formal proof, you can also show that whenever one expression is true, the other must be true (and vice versa) using logical implications.

Let’s apply the methods above to your specific expressions.

Method 1: Truth Table #

First, we’ll list all 8 possible truth value combinations for A, B, and C, then compute both expressions for each case:

Every row has identical results for both expressions—so they’re definitely equivalent.

Method 2: Boolean Algebra #

We’ll use the distributive law of boolean algebra, which works in two directions:

• X && (Y || Z) = (X && Y) || (X && Z) (expansion)
• (X && Y) || (X && Z) = X && (Y || Z) (factorization, reverse of expansion)

Starting with the first expression:

(A && C) || (B && C)

Notice C is a common term in both parts of the OR. We can factor out C using the reverse distributive law:

= C && (A || B)

Since boolean AND is commutative (X && Y = Y && X), we can rearrange this to:

= (A || B) && C

That’s exactly the second expression! By transforming one into the other using standard boolean rules, we’ve proven their equivalence.

Method 3: Plain-Language Reasoning #

Let’s translate both expressions into everyday terms:

• (A && C) || (B && C): "Either both A and C are true, or both B and C are true." For this to hold, C must be true, and at least one of A or B also has to be true.
• (A || B) && C: "At least one of A or B is true, AND C is true." This describes the exact same scenario—C is true, and either A or B (or both) is true.

No matter how you slice it, these two expressions mean the same thing.

内容的提问来源于stack exchange，提问作者jchi2241