Using Boolean laws simplify each of the following propositions:a. a ∧ b ∨ a ∧ ¬b

To simplify the proposition a ∧ b ∨ a ∧ ¬b using Boolean laws, we can follow these steps:

1. Distributive Law: Apply the distributive law to the proposition. This law states that a ∧ (b ∨ c) is equivalent to (a ∧ b) ∨ (a ∧ c).

   So, we can rewrite the proposition as (a ∧ b) ∨ (a ∧ ¬b).

2. Factor out a: Notice that both terms in the proposition have a common factor of 'a'. We can factor out 'a' from both terms.

   This gives us a ∧ (b ∨ ¬b).

3. Law of Excluded Middle: According to the law of excluded middle, b ∨ ¬b is always true. This means that the proposition b ∨ ¬b is equivalent to the constant 'true'.

   Therefore, we can simplify the proposition to a ∧ true.

4. Identity Law: The identity law states that a ∧ true is equivalent to 'a'.

   Finally, we can simplify the proposition to just 'a'.