4. computer maths and logic 4.2 boolean logic 4.2.3 simplifying boolean expressions
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4. Computer Maths and Logic
4.2 Boolean Logic
4.2.3 Simplifying Boolean Expressions
Complex expressions can be constructed using the operators e.g. (A B) • (C + D), which is equivalent to writing (A xor not B) and (neither C nor D nor both).
In the exam, you will not have to consider more than 3 inputs
Complex Expressions
To evaluate a complex Boolean expression, break it down to smaller parts then use a truth table e.g. firstly (A B):
Complex Expressions
1011
0101
0010
1100
(A B)BBA
OutputIntermediate
Inputs
Complex Expressions
There’s a worksheet to help you evaluate the rest of the expression
Draw similar truth tables for A + B and A • B.
Truth tables can be used as a means of checking if two expressions are equivalent.
Complex Expressions
Complex expressions can often be reduced to simpler ones
This is similar to work you have done in algebra in maths
Look out for the following expressions which are always true
Simplifying Expressions
A • 0 = 0 A + 1 = 1
Simplifying Expressions
A • 1 = A A + 0 = A
Simplifying Expressions
A + A = A A • A = A
Simplifying Expressions
A • B = B • A A + B = B + A (the commutative law)
Simplifying Expressions
A • (B • C) = (A • B) • C = A • B • C
(the associative law)
Simplifying Expressions
A + (B + C) = (A + B) + C = A + B + C
(the associative law)
Simplifying Expressions
A • (B + C) = A•B + A•C
(A + B) • (A + C) = A•A + A•C + B•A + B•C
(the distributive law) Verify these with truth tables
Simplifying Expressions
A + A = 1 A • A = 0
(De Max's laws)
Simplifying Expressions
A + B = A • B A + B = A • B
De Morgen's lawVerify these with truth tables
Simplifying Expressions
anything in brackets is done first
• is done before +
Simplifying Expressions