simplify the following expression with the help of XOR and XNOR:

X = ABCD+A(~B)(~C)D+(~A)(~B)

To simplify the given expression using XOR (exclusive OR) and XNOR (exclusive NOR), we need to apply the following properties:

1. XOR: A XOR B = (A AND ~B) OR (~A AND B)
2. XNOR: A XNOR B = (A AND B) OR (~A AND ~B)

Let's simplify the expression step by step:

X = ABCD + A(~B)(~C)D + (~A)(~B)

First, let's distribute the terms inside parentheses:

X = ABCD + A(~B)(~C)D + (~A)(~B)
= ABCD + A(~B)D(~C) + (~A)(~B)

Now, let's apply the XOR property to the second term:

~B XOR ~C = (~B AND ~C) OR (B AND C)

X = ABCD + A(~B)D(~C) + (~A)(~B)
= ABCD + A[(~B AND ~C) OR (B AND C)]D + (~A)(~B)

Now let's simplify the third term using the XNOR property:

~A XNOR ~B = (~A AND ~B) OR (A AND B)

X = ABCD + A[(~B AND ~C) OR (B AND C)]D + (~A)(~B XNOR B)

Finally, let's simplify the XNOR term:

~B XNOR B = (~B AND B) OR (B AND ~B) = FALSE OR FALSE = FALSE

Therefore, the expression simplifies to:

X = ABCD + A[(~B AND ~C) OR (B AND C)]D + (~A)(FALSE)

Since FALSE is equivalent to 0, the last term can be eliminated:

X = ABCD + A[(~B AND ~C) OR (B AND C)]D

Thus, the simplified expression is X = ABCD + A[(~B AND ~C) OR (B AND C)]D.