Correct : b,d

Explanation:
1. Understand the Boolean Function X(a, b, c):
The function X produces a '1' when at least two of its three inputs are '1'. This is the exact definition of a Majority Function, which can be algebraically written as:
    X(a, b, c) = ab + bc + ca

2. Evaluate Statement (b): X(a, b, X(a, b, c)) = X(a, b, c)
Let's substitute the algebraic expression of the majority function:
• Left-Hand Side (LHS) = ab + b·X(a, b, c) + a·X(a, b, c)
• Factoring out X(a, b, c): LHS = ab + (a + b)·X(a, b, c)
• Substitute X(a, b, c) = ab + bc + ca:
    LHS = ab + (a + b)(ab + bc + ca)
    LHS = ab + a²b + abc + a²c + ab² + b²c + abc
• Since a² = a, b² = b, and tracking absorbing terms (like ab + ab = ab):
    LHS = ab + ab + abc + ac + ab + bc + abc
    LHS = ab + bc + ca = X(a, b, c)
• Since LHS equals RHS, Statement (b) is Correct.

3. Evaluate Statement (d): X(a, b, c) = X(a, X(a, b, c), X(a, c, c))
Let's simplify the inner components on the Right-Hand Side (RHS):
• For X(a, c, c): At least two inputs must be '1'. Since two inputs are identical ('c'), it outputs '1' if and only if c = 1, regardless of 'a' (because if c=1, two inputs are already 1). Thus, X(a, c, c) simplifies simply to c.
• Now substitute this back into the expression:
    RHS = X(a, X(a, b, c), c)
• Let $M = X(a, b, c)$. The expression expands to the majority of {a, M, c}:
    RHS = a·M + M·c + c·a = ac + M(a + c)
• Substitute $M = ab + bc + ca$:
    RHS = ac + (ab + bc + ca)(a + c)
    RHS = ac + a²b + abc + a²c + abc + bc² + ca²
    RHS = ac + ab + abc + ac + abc + bc + ac
• Combining the redundant terms via absorption laws leaves us with:
    RHS = ab + bc + ca = X(a, b, c)
• Since RHS equals LHS, Statement (d) is Correct.

4. Why Statements (a) and (c) are Incorrect:
• Statement (a): The majority function is not associative. For instance, if a=1, b=1, c=0, d=0, e=0, the LHS yields 0 while the RHS yields 1.
• Statement (c): The LHS expands to a complex boolean polynomial, while the RHS simplifies strictly to $a \cdot (c + d)$ which does not match the coverage logic of the LHS.