Simplifying Logical Expressions [SOLVED]

In summary, the conversation discusses the simplification of a Boolean expression using DeMorgan's Law and the Distributive Property. The original expression is NOT(a < 20 AND (b < 10 OR b > 10)), which can be simplified to (NOT a < 20 OR NOT b < 10) AND (NOT a < 20 OR NOT b > 10). The question arises whether to use DeMorgan's Law again, which results in the simplified form of a >= 20 OR b = 10. However, it is noted that this may not always be the most simplified form, as in the case of (NOT a < 20 AND NOT (b <= 10 OR b >= 15)), which
  • #1
Valtham
3
0
[SOLVED] Simplifying Expressions

Hello all I am a bit confused about the laws I am learning about right now and attempted some of my own exercises to understand them a bit.

I have the problem NOT(a < 20 AND (b < 10 OR b > 10)) and I need to simplify it.

From my understanding I can use DeMorgan's Law which then gives me the expression NOT a < 20 OR NOT(b < 10 OR b > 10). I can then use the Distributive Property to create the expression (NOT a < 20 OR NOT b < 10) AND (NOT a < 20 OR NOT b > 10).

What I am confused about is after I use DeMorgan's Law the first time is it correct to use the Distributive Law next? Or should I have used DeMorgan's Law again?
 
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  • #2
Valtham said:
What I am confused about is after I use DeMorgan's Law the first time is it correct to use the Distributive Law next? Or should I have used DeMorgan's Law again?
You can't use the distributive law immediately after the first De Morgan's law. Distributivity requires that the expression has both a disjunction and a conjunction. So, first you need to convert NOT(b < 10 OR b > 10) into (NOT b < 10) AND (NOT b > 10) and then use distributivity. You indeed get (NOT a < 20 OR NOT b < 10) AND (NOT a < 20 OR NOT b > 10).

If < denoted the regular order, then I believe the simplest form of this is expression is a >= 20 OR b = 10.
 
  • #3
Thanks for the reply. Makes total sense that I would have to use DeMorgan's Law again. I can see how you get a >= 20 OR b = 10, but what if the expression had been (NOT a < 20 AND NOT (b <= 10 OR b >= 15)). Using that expression the "simplified" version would be a >= 20 OR b >= 10 AND b <= 15. To me that expression hardly seems simplified, and all we did was remove the "NOT" connectives pretty much. What does "simplifying an expression" mean exactly? We weren't really given a precise definition other than an example that removes as many variables and connectives as possible.
 
  • #4
Valtham said:
what if the expression had been (NOT a < 20 AND NOT (b <= 10 OR b >= 15)). Using that expression the "simplified" version would be a >= 20 OR b >= 10 AND b <= 15.
It should say, "... b > 10 AND b < 15." Also, usually AND is considered to have higher priority than OR, so omitting parentheses is OK, but unless this is an explicit convention in your course, it may still make sense to put parentheses around b > 10 AND b < 15 to remove any ambiguity.

Valtham said:
To me that expression hardly seems simplified, and all we did was remove the "NOT" connectives pretty much. What does "simplifying an expression" mean exactly? We weren't really given a precise definition other than an example that removes as many variables and connectives as possible.
There are different measures with respect to which simplification can be defined. Here the answer has 2 connectives vs 4 in the original expression, so in this sense it is simpler. On the other hand, the number of atomic propositions is the same. Informally, for me it is a little easier to understand a >= 20 than NOT a < 20. Also, b > 10 AND b < 15 is often abbreviated as 10 < b < 15, which makes it even simpler.
 
  • #5


I would say that your approach is correct. DeMorgan's Law allows for the simplification of logical expressions by negating the entire expression and then applying the opposite logical operator. In this case, you correctly applied DeMorgan's Law to the first part of the expression, resulting in NOT a < 20 OR NOT(b < 10 OR b > 10).

Next, the Distributive Property can be used to simplify the expression even further by distributing the negation to each term within the parentheses. This results in (NOT a < 20 OR NOT b < 10) AND (NOT a < 20 OR NOT b > 10).

So, in summary, your approach is correct and you can continue to simplify the expression using other laws and properties as needed. Keep up the good work!
 

FAQ: Simplifying Logical Expressions [SOLVED]

What does it mean to "simplify" an expression?

Simplifying an expression means to rewrite it in a shorter, more compact form while still maintaining its equivalent value. This is often done by combining like terms, factoring, and using the order of operations.

How do I know when an expression is fully simplified?

An expression is fully simplified when there are no more like terms to combine, no more parentheses to simplify, and all operations have been performed according to the order of operations.

Can I simplify an expression with variables?

Yes, expressions with variables can also be simplified by combining like terms and following the order of operations. However, the final result may still contain variables.

Are there any rules for simplifying expressions?

Yes, there are a few rules to keep in mind when simplifying expressions. These include the distributive property, the commutative and associative properties, and the order of operations.

Why is simplifying expressions important?

Simplifying expressions allows us to manipulate and solve equations more easily, making it a crucial skill in algebra and other areas of mathematics. It also helps us to identify patterns and relationships between different expressions.

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