1. Apply De Morgan's theorems to simplify the expression:2. Simplify the expression F = A B D+A B' using Boolean identities.3. Reduce A (A + B) to the least number of terms.4. Simplify the expression F=A B ' D+A B' D'Short answer questions:1. Simplify the following expressions using boolean laws.𝐹 = 𝐴. 𝐵 + 𝐴 . (𝐶. 𝐷 + 𝐶. 𝐷 ) 𝐹 = 𝐴. 𝐵. 𝐶 + 𝐴 + 𝐴. 𝐶. 𝐵Long answer questions:𝐴 + 𝐵 . (𝐶 + 𝐷)F = 𝐴.(A+B)+𝐴. 𝐵

Sure, let's go step by step:

1. De Morgan's theorems state that:

   • The complement of the sum of two variables is equal to the product of their complements. (A + B)' = A' . B'
   • The complement of the product of two variables is equal to the sum of their complements. (A . B)' = A' + B'

   If you have a specific expression, I can apply these theorems to it.

2. Simplify the expression F = A B D + A B':

   • Using the Consensus theorem (AB + A'C + BC = AB + A'C), we can simplify it to F = A B D + A B' D + A B' D' = A B' + A B D

3. Reduce A (A + B) to the least number of terms:

   • Using the Absorption law (A + AB = A), we can simplify it to A.

4. Simplify the expression F = A B' D + A B' D':

   • Using the Consensus theorem, we can simplify it to F = A B' D + A B' = A B'

Short answer questions:

1. Simplify the following expressions using boolean laws.
   • 𝐹 = 𝐴. 𝐵 + 𝐴 . (𝐶. 𝐷 + 𝐶. 𝐷 ) = A.B + A.C (using Absorption law)
   • 𝐹 = 𝐴. 𝐵. 𝐶 + 𝐴 + 𝐴. 𝐶. 𝐵 = A (using Absorption law)

Long answer questions:

• 𝐴 + 𝐵 . (𝐶 + 𝐷) = A + B.C + B.D (using Distributive law)
• F = 𝐴.(A+B)+𝐴. 𝐵 = A + A.B = A (using Absorption law)