View source for A-level Computing/AQA/Paper 2/Fundamentals of computer systems/Building circuits

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{{DISPLAYTITLE:Fundamental Hardware Elements of Computers: Building circuits}}
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{{CPTAmbox|text=Sometimes you might get an ambiguous equation such as <math>A.B+C</math>.  Do you work out the AND or the OR first?  The rule is to treat the AND part first so we would treat <math>A.B+C = (A.B)+C</math>}}
A common question in the exam is to be given some boolean algebra and be asked to express it as logic gates.  Let's take a look at an addition and subtraction example that you should be familiar with:
 <math>9-(7+1)</math>
First we are going to deal with the inner-most brackets
 <math>(7+1) = 8</math>
Finally we combine this answer with the <math>9-</math>
 <math>= 9-(8) = 1</math>
It will work exactly in the same way for boolean algebra, but instead of using numbers to store our results, we'll use logic gates:
{{CPTExample|title=Example: Building circuits}}
 <math>C.(A+B)</math>
As with any equation, we are going to deal with the inner-most brackets first<math>(A+B)</math>, then combine this answer with the <math>C.</math><br/>
[[File:CPT-logic-gate building circuit.svg]]
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{{CPTExercise|title=Exercise: Building circuits}}
{{CPTQuestion|Draw the circuit diagrams for the following. (Remember: do we deal with AND or OR first?):}}
{{CPTQuestion|<math>A.B+C</math>}}
{{CPTAnswer|[[File:CPT-logic-gate conversion (A.B)+C.svg]]}}
{{CPTQuestion|<math>(\overline{A}+\overline{B}).C</math>}}
{{CPTAnswer|[[File:CPT-logic-gate conversion (-A+-B).C.svg]]}}
{{CPTQuestion|<math>\overline{(A+B)}.C</math>}}
{{CPTAnswer|[[File:CPT-logic-gate conversion ¬(A.+B).C.svg]]}}
{{CPTQuestion|<math>(A.\overline{B})\oplus(B.C)</math>}}
{{CPTAnswer|[[File:CPT-logic-gate conversion (A.¬B)(+)(B.C).svg]]}}
{{CPTQuestion|<math>\overline{(A.B).(C+\overline{D})}</math>}}
{{CPTAnswer|[[File:CPT-logic-gate conversion ¬((A.B).(C+¬D)).svg]]}}
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A common question in the exam is to give you a description of a system.  You'll then be asked to create a boolean statement from this description, and finally build a logic gate circuit to show this system:
{{CPTExample|title=Example: Building circuits}}
Using boolean algebra describe the following scenario:
{{cquote|A car alarm is set off if a window is broken or if it senses something moving inside car, and the car is not being towed, or the engine is not on.}}
Where:
* A = being towed,
* B = window broken,
* C = engine on,
* D = senses movement
Before you rush into answering a question like this, let's try and break it down into its components.  The questioner will often be trying to trick you.  The two occasions that the alarm will sound are:
<math>B+D</math> but there is a caveat, the alarm will sound if either of these are true '''AND''' two things are also true, namely the engine is '''NOT''' on, and the car is '''NOT''' being towed:
<math>\overline{A}+\overline{C}</math>

Combining both we get (remember the brackets!):
<math>(B+D).(\overline{A}+\overline{C})</math>

The next step is to create a diagram out of this:
[[File:CPT-logic-gate conversion (B+D).(¬A+¬B).svg]]

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{{CPTExercise|title=Exercise: Building circuits}}
{{CPTQuestion|1=A security system allows people of two different clearance levels access to a building. Either they have low privileges and they have a card and they are not carrying a mobile. Alternatively they have a key and are allowed to carry a mobile.

The inputs available are:
* A = carrying a card
* B = carrying a mobile phone
* C = carrying a key
Write down the boolean equation to express this:}}
{{CPTAnswer|<math>(A.\overline{B})+(C)</math>

If you wrote: <math>(A.\overline{B})+(C.B)</math> You'd be wrong!  The reason being the text says: ''Alternatively they have a key and are <u>allowed</u> to carry a mobile.''  This doesn't mean <math>C.B</math>, it means they can carry a mobile, or they can choose not to: <math>C.(B+\overline{B})</math>, which simplifies to: <math>C.(B+\overline{B}) = C.(1) = C</math>.}}
{{CPTQuestion|Draw the logic gate diagram to solve this:}}
{{CPTAnswer|[[File:CPT-logic-gate conversion (A.¬B)+(C).svg]]}}
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