# Given the phrase “Where NONE of the following are TRUE” and two statements how should a boolean logic be composed?

Let's have two statements

1. (value > 10)
2. (value < 25)

And a list of items with the following values

• 10
• 20
• 30

This is what a truth table would give

Item  Value  (value > 10)  (value < 25)
----  -----  ------------  ------------
1     10     FALSE         TRUE
2     20     TRUE          TRUE
3     30     TRUE          FALSE


## Example 1

Where ALL of the following are TRUE
value > 10
value < 25


This one is easy and we get the following

Where (value > 10) AND (value < 25)


The result is then a single value of 20

## Example 2

Where NONE of the following are TRUE
value > 10
value < 25


This is where I am not sure of what to generate.

This would be "simple" as it is only a NOT of the whole expression

Where NOT ((value > 10) AND (value < 25))


However, the result is then two values (10 and 30)

From what someone would think of NONE of the two statements would be something like:

Where NOT ((value > 10) OR (value < 25))


And the result would be that no items are produced.

What is the correct meaning of NONE here?

• The correct meaning is "Where (NOT (value > 10)) AND (NOT (value < 25))". Applying De Morgan's laws we get: "Where NOT ( (NOT(NOT(value > 10))) OR (NOT(NOT(value < 25))) )" and the equivalent result is "Where NOT ((value > 10) OR (value < 25))" :-) – Vor Oct 18 '12 at 21:38
• @vor, make an answer out of it.. – Ran G. Oct 19 '12 at 2:55
• @vor: Yepp, this is an answer. Please post it as answer. – A.Schulz Oct 19 '12 at 7:29

The correct meaning is:

"Where (NOT (value > 10)) AND (NOT (value < 25))".

Applying one of the De Morgan's laws:

1) P AND Q <=> NOT( (NOT P) OR (NOT Q) )
2) P OR Q <=> NOT( (NOT P) AND (NOT Q) )

we get:

"Where NOT ( (NOT(NOT(value > 10))) OR (NOT(NOT(value < 25))) )"

and the equivalent final result is

"Where NOT ((value > 10) OR (value < 25))" :-)