Boolean algebra is a fundamental mathematical system that works with two values, typically represented as true (1) and false (0). It is essential in digital electronics and computer science for designing and analyzing logical circuits and expressions. Understanding the various laws of Boolean algebra helps in simplifying complex logical statements and optimizing digital systems.

The question presents two specific laws and asks us to identify their type:

• Law 1: \(A + A \cdot B = A\)
• Law 2: \(A(A + B) = A\)

Absorption Laws in Boolean Algebra

The laws provided are prime examples of the Absorption Laws in Boolean algebra. These laws are used to simplify Boolean expressions where one variable or term is "absorbed" by another. They are very useful for reducing the complexity of logical circuits and expressions.

The two forms of the absorption law are:

• The OR form: \(A + A \cdot B = A\)
• The AND form: \(A \cdot (A + B) = A\)

Let's delve into the proofs of these laws to understand how they work.

Proving Law 1: \(A + A \cdot B = A\)

This law states that if we take a variable \(A\) and perform an OR operation with the logical product (AND) of \(A\) and another variable \(B\), the result is simply \(A\). We can prove this using other basic Boolean algebra principles:

• Begin with the left-hand side of the equation: \(A + A \cdot B\)
• Using the Identity Law (\(X = X \cdot 1\)), we can rewrite \(A\) as \(A \cdot 1\): \(A \cdot 1 + A \cdot B\)
• Apply the Distributive Law (\(X \cdot Y + X \cdot Z = X \cdot (Y + Z)\)). Here, \(X\) is \(A\), \(Y\) is \(1\), and \(Z\) is \(B\): \(A \cdot (1 + B)\)
• According to the Null Law (also known as Dominance Law), \(1 + X = 1\) for any variable \(X\). So, \(1 + B = 1\): \(A \cdot 1\)
• Finally, apply the Identity Law again (\(X \cdot 1 = X\)): \(A\)

Therefore, the law \(A + A \cdot B = A\) is proven.

This law can also be verified using a truth table, which lists all possible combinations of input values and their corresponding output values:

As observed from the truth table, the column for \(A + A \cdot B\) is exactly the same as the column for \(A\), which confirms the validity of this absorption law.

Proving Law 2: \(A(A + B) = A\)

This law states that if we perform an AND operation between a variable \(A\) and the logical sum (OR) of \(A\) and another variable \(B\), the result is simply \(A\). Let's prove this step-by-step:

• Start with the left-hand side of the equation: \(A(A + B)\)
• Apply the Distributive Law (\(X \cdot (Y + Z) = X \cdot Y + X \cdot Z\)). Here, \(X\) is \(A\), \(Y\) is \(A\), and \(Z\) is \(B\): \(A \cdot A + A \cdot B\)
• According to the Idempotent Law (\(X \cdot X = X\)), \(A \cdot A = A\): \(A + A \cdot B\)
• Notice that the expression \(A + A \cdot B\) is precisely the first absorption law, which we have already proven to be equal to \(A\).

Hence, the law \(A(A + B) = A\) is proven.

This law can also be verified using a truth table:

The truth table clearly shows that the column for \(A(A + B)\) is identical to the column for \(A\), thus confirming this absorption law.

Other Boolean Laws Compared

It's helpful to briefly review why other types of Boolean laws do not apply to the given expressions:

• Double Negation Laws: These laws deal with the negation of a negated variable, for example, \((A')' = A\). The laws in the question do not involve any negation.
• Distributive Laws: These laws describe how AND distributes over OR, or vice-versa, such as \(A \cdot (B + C) = A \cdot B + A \cdot C\) or \(A + (B \cdot C) = (A + B) \cdot (A + C)\). While distributive law is used as a step in proving absorption laws, the given expressions themselves are not examples of the distributive law.
• Associative Laws: These laws state that the grouping of variables does not affect the outcome for operations like OR or AND, for instance, \(A + (B + C) = (A + B) + C\) or \(A \cdot (B \cdot C) = (A \cdot B) \cdot C\). The given laws are simplification rules, not about grouping.

Therefore, based on their properties and proofs, the laws \(A + A \cdot B = A\) and \(A(A + B) = A\) are correctly identified as Absorption Laws.