Simplify the boolean equation given below?

The expression simplifies to 1 (always true). By De Morgan's law, (A + B)' = A'B'. Then A'B' + AB + A can be regrouped as A'B' + A(B + 1). Since (B + 1) = 1, we get A'B' + A = 1 because either A is true (making the result 1) or A is false (making A'B' = 1 when B is also false, which covers all cases).

Step-by-step Derivation:
Step 1: Apply De Morgan's law to (A + B)'
(A + B)' = A'B'

Step 2: Substitute back into the expression
Y = A'B' + AB + A

Step 3: Factor out A from the last two terms
Y = A'B' + A(B + 1)

Step 4: Simplify B + 1 (OR with 1 is always 1)
Y = A'B' + A(1)
Y = A'B' + A

Step 5: Use the absorption-like principle: A'B' + A
When A = 1: Y = 0 + 1 = 1
When A = 0: Y = 1·B' + 0 = B'

Step 6: Re-examine using consensus or truth table
A | B | A'B' | AB | A | Y
0 | 0 | 1 | 0 | 0 | 1
0 | 1 | 0 | 0 | 0 | 0 ← Wait, let me recalculate

Step 6 (Corrected): Full truth table evaluation
A | B | (A+B)' | AB | A | (A+B)'+AB+A
0 | 0 | 1 | 0 | 0 | 1
0 | 1 | 0 | 0 | 0 | 0 ← This gives 0, not 1
1 | 0 | 0 | 0 | 1 | 1
1 | 1 | 0 | 1 | 1 | 1

Step 7: Re-analysis - The result is NOT always 1
Looking at row 2: when A=0, B=1, Y=0
So the answer is NOT C.

Step 8: Simplify correctly using Boolean algebra
Y = A'B' + AB + A
Y = A'B' + A(B + 1)
Y = A'B' + A
Y = (A' + A)(B' + A) [using distribution in reverse]
Y = 1 · (A + B')
Y = A + B'

Actually, let me use the correct method:
Y = A'B' + AB + A
Y = A'B' + A(B + 1)
Y = A'B' + A

Using X + X'Y = X + Y:
Y = A + A'B' = A + B' (when we apply the consensus theorem correctly)

Wait - truth table check:
A=0, B=0: 1 + 0 + 0 = 1 ✓ | A + B' = 0 + 1 = 1 ✓
A=0, B=1: 0 + 0 + 0 = 0 ✓ | A + B' = 0 + 0 = 0 ✓
A=1, B=0: 0 + 0 + 1 = 1 ✓ | A + B' = 1 + 1 = 1 ✓
A=1, B=1: 0 + 1 + 1 = 1 ✓ | A + B' = 1 + 0 = 1 ✓

So Y = A + B' is the correct simplification, which matches NEITHER of the given options exactly. However, reviewing the original options and given that none match A + B', let me verify if there's an error in the problem or if I should reconsider.

Given the available options and the requirement to choose from them: The expression does NOT simplify to 1, A', or B' universally. However, examining standard simplification: Y = A'B' + AB + A should yield A + B' by consensus theorem.