Logic and distributive laws

In summary: For example, in the second step, you could have said:(\neg R\land \neg S) \lor CThis will give you the same result as the distributive law, but it is not the correct way to solve the problem.
  • #1
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Im just reading this one example and i am stumped at this one step.

[tex] (R\to C) \land (S \to C) \\
(\neg R\lor C) \land (\neg S \lor C) \ \ \ \ \ \textrm{by conditional law}\\
(\neg R\land \neg S) \lor C \ \ \ \ \textrm{by distributive law}[/tex]

I don't understand how it went from the second step to the third

my attempt from the second step was:
[tex](\neg R \land \neg S) \lor (\neg R \land C) \lor (C \land \neg S) \lor C[/tex]
but don't know where to go from here.

Did I do correctly applied the distributive law?
 
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  • #2
The distributive laws in the propositional calculus say that

A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C)​

and also what you get when the symbols ∧ and ∨ are interchanged:

A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C).​

The A, B, C can of course stand for any propositions at all. Now you can probably see how choosing the appropriate A, B, C from your problem will get you from step 2 to step 3.
 
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  • #3
The Subject said:
Im just reading this one example and i am stumped at this one step.

[tex] (R\to C) \land (S \to C) \\
(\neg R\lor C) \land (\neg S \lor C) \ \ \ \ \ \textrm{by conditional law}\\
(\neg R\land \neg S) \lor C \ \ \ \ \textrm{by distributive law}[/tex]

I don't understand how it went from the second step to the third
They could have said "by distributive law used in reverse", but actually there is no "forward" and "reverse" direction to the distributive law. It a human tendency to think what we go from the left hand side of an equivalence to the right right hand side, but you don't have to use equivalences that way. Do you see how to go from the third line back to the second line using the distributive law?

my attempt from the second step was:
[tex](\neg R \land \neg S) \lor (\neg R \land C) \lor (C \land \neg S) \lor C[/tex]
but don't know where to go from here.

I think you tried to do more than apply the distributive law to the second step.
 
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1. What is the purpose of logic and distributive laws?

Logic and distributive laws are used to identify and organize patterns and relationships between mathematical statements. They help to simplify and make complex mathematical expressions more manageable.

2. How do distributive laws work?

Distributive laws allow us to expand or simplify mathematical expressions that involve multiplication and addition or subtraction. They state that when a common factor is present in both terms being added or subtracted, the factor can be factored out and the operation can be performed on the remaining terms.

3. What is the difference between distributive laws and associative laws?

Distributive laws and associative laws both deal with the relationships between mathematical operations, but they serve different purposes. Distributive laws help to simplify expressions by distributing a common factor, while associative laws allow us to change the grouping of terms without changing the result.

4. How are distributive laws used in real-world applications?

Distributive laws are used in many real-world applications, such as in finance, engineering, and computer science. They are used to simplify and manipulate equations to solve problems and make processes more efficient.

5. Are there any limitations to using distributive laws?

While distributive laws are useful for simplifying and manipulating expressions, they do have limitations. They can only be applied to expressions involving addition and multiplication, and they cannot be used if the common factor is being subtracted or divided. Additionally, they may not always result in the most simplified form of an expression.

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