Let $X$ be a 3-variable Boolean function that produces output as '1' when at least two of the input variables are ‘1'. Which of the following statement(s) is/are CORRECT, where $a, b, c, d, e$ are Boolean variables?
To solve the given problem, we need to analyze which Boolean expressions satisfy the given function and equivalences.
The Boolean function \(X(a, b, c)\) returns '1' if at least two of the input variables are '1'. Understanding this behavior will guide us in evaluating each statement.
Let's evaluate each option:
- \(X(a, b, X(c, d, e)) = X(X(a, b, c), d, e)\):
- Analyze when \(X(c, d, e)\) is 1: It is 1 if at least two of the variables \(c, d, e\) are 1.
- The left side becomes 1 if at least two among \(\{a, b, X(c, d, e)\}\) are 1.
- The right side depends on values of \(a, b, c\): It is 1 if at least two among \(\{X(a, b, c), d, e\}\) are 1.
- This statement is generally not true as the conditions to become 1 may differ.
- \(X(a, b, X(a, b, c)) = X(a, b, c)\):
- By definition of \(X(a, b, c)\), this holds true as the addition of \(X(a, b, c)\) doesn't change the result, as the function checks for at least two 1s in either case.
- \(X(a, b, X (a, c, d)) = (X(a,b,a) \text{ AND } X(c, d, c))\):
- For the left side, \(X(a, b, X(a, c, d))\) becomes 1 if at least two among \(\{a, b, X(a, c, d)\}\) are 1.
- For the right side, \(X(a, b, a) = a\) and \(X(c, d, c) = c \lor d\).
- The resultant condition required for equality is not satisfied generally, this is not a correct expression.
- \(X(a, b, c) = X(a, X(a, b, c), X(a, c, c))\):
- Simplify: \(X(a, b, c)\) is 1 if at least two of \(\{a, b, c\}\) are 1.
- For the right-hand side:
- \(X(a, b, c)\) is 1 if \(\{a, b, c\}\) have at least two 1s.
- \(X(a, c, c)\) is 1 if \(a \land c\) are 1, i.e., simple pair check.
- Therefore, \(X(a, X(a, b, c), X(a, c, c))\) effectively preserves the logic of having at least two 1s overall.
- Both sides evaluate to 1 under corresponding conditions; thus this statement is Correct.
Thus, the Correct answers are:
- \(X(a, b, X(a, b, c)) = X(a, b, c)\)
- \(X(a, b, c) = X(a, X(a, b, c), X(a, c, c))\)
Important Questions from Boolean Algebra
How many different Boolean functions of degree n are there?
Consider a Boolean function of ‘n’ variables. The order of an algorithm that determines whether the Boolean function produces a output 1 is:
The marginal probability of cavity P(cavity) is ________.
The probability of a cavity, given evidence of a toothache, P(cavity | toothache) is _________.
The probability of a toothache, given evidence of a cavity, P(toothache | cavity) is ________.
