# Boolean Algebra Simplification Laws with Proof

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We now discuss a few basic laws used in logic
simplification. Each of these basic laws is stated along with its complementary
law. It may be seen that these laws can be proved using either truth tables or the
basic rules given above.

**2.16.1**

**Idempotent Laws**

Idempotent laws
state that :

*x + x = x*

*(2.14a)*

*x*

*×**x = x*(2.14b)

The word idempotent (same power) was
coined by the American professor of computer science Benjamin C. Pierce. It was
derived from the words

*idem*(same) and*potent*(power). Proofs of the relations stated above are given in Table 2.10. Inspection of columns 3 and 4 of Table 2.10 reveals that the summation*x + x*and the product*xx*results in*x*itself.

**Table 2.10**Truth table proving**Eqs. (2.14a) and (2.14b)**x |
x |
x + x = x |
x× x = x |

01 |
01 |
01 |
01 |

**2.16.2**

**Absorption Laws**

Let

*x*and*y*be Boolean variables. Then, absorption laws state that:

*x + xy*=

*x*(2.15a)

*x*(

*x + y*)

*=*

*y*(2.15b)

**Proof of absorption laws using truth table**

**:**As stated above, these laws can be proved by using truth tables, or otherwise. Proof of the first of the two absorption laws (i.e.,

*x + x y*=

*x*) is given in Table 2.11. Similar steps can be used to prove the remaining relations. The first and last columns of Table 2.11 agree showing the validity of the proof. Here, the variable

*y*is absorbed by the variable

*x*, and hence the name

*absorption law*

**.**

**Table 2.11**Truth table to prove

*x + xy*=

*x*

x |
Y |
x y |
x + x y |

0011 |
0101 |
0001 |
0011 |

**Proof of Absorption law using algebraic method:**

**We can prove the first of the absorption laws by using basic algebra also. For this, we write the LHS of the given equation:**

LHS =

*x + x y = x*(**1**+*y*) =*x∙***1**=*x*= RHS_{}
where we have used the basic rule

**1**+*y*=**1**. It can be seen that this proof is comparatively faster. However, it may be simpler and faster only for equations containing a smaller number of variables.**Proof of the complement of the absorption law:**LHS of

**Eq. (2.15b) is given by**

*LHS =*

*x*(

*x + y*)

*= x*+

*x y*=

*x*(

**1**+

*y*) =

*x*= RHS

where for proving the law, we have used two basic
relations, viz.,

*x x*=*x*, and 1 +*y*=**1**.**2.16.3 Laws Related to Absorption Laws**

There are a few other laws associated with the
absorption law. These are discussed below.

**(a) Subsidiary 1:**This law states that

*x*+

*x′y*=

*x*+

*y*(2.16a)

**Proof:**The LHS of the given expression may be written in the form:

LHS =

*x*+*x′y*=*x*(1+*y*)+*x′y*=*x*+*xy*+*x′y*=*x*+*y*(*x*+*x′*) =*x*+*y*= RHS
where we have introduced the term

**1**+*y*=**1**into the LHS part of the equation for the simplification.**(b) Subsidiary 2 (Complementary**

**of law a):**

**This law states that:**

*x*(

*x′*+

*y*)

*=*

*xy*(2.16b)

**Proof:**The LHS of the given expression may be written in the form:

LHS =

*x*(*x′*+*y*)*=**x**x′ +**xy**=**xy*= RHS**(c) Subsidiary 3:**

**This law states that**

*x′ +*

*xy*

*=*

*x′*+

*y*

*(2.17a)*

**Proof:**The LHS of the given expression may be written in the form:

LHS =

*x′*+*xy*=*x′*(1+y)+*xy*=*x**′*+*x′**y*+*xy*=*x′*+*y*(*1*+*x*) =*x′*+*y*= RHS**(d) Subsidiary 4 (Complementary**

**of law c):**

**This law states that**

*x′*(x+y) =

*x′*

*y*

*(2.17b)*

**Proof:**The LHS of the given expression may be written in the form:

LHS =

*x′*(x+y) =*x′x**+**x′**y**=**x′**y*= RHS**2.16.4 Commutative Laws**

Commutative laws state that:

*x + y = y + x*

*(2.18a)*

*x y = y x*(2.18b)

**2.16.5 Associative Laws**

Associative laws state that:

*x +*(

*y + z*)

*=*(

*x + y*)

*+ z*

*(*2.19a)

*x*(

*y z*)

*=*(

*x y*)

*z*

*(2.19b)*

**2.16.6**

**Distributive Laws**

Distributive laws state that:

*x*(

*y + z*)

*= x y + x z*

*(2.20a)*

*x + yz =*(

*x + y*) (

*x +z*) (2.20b)

The second law given by Eq. (2.20b), viz.,

*x + y z =*(*x + y*)(*x +z*) is not found to conventional algebra and is special of Boolean algebra alone.
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