Question

which of the following options best completes the column for (y && y) || (x && !x) in the truth table? x y (y && y) (x && !x) (y && y) || (x && !x) t t t t f f f t ? f f

Explanation:

Step1: Analyze (y && y)

When \(y = T\), \(y\ && y=T\ && T = T\). When \(y = F\), \(y\ && y=F\ && F = F\).

Step2: Analyze (x &&!x)

For any \(x\), \(x\) and \(!x\) cannot be both true. So \(x\ &&!x = F\) always.

Step3: Analyze (y && y) || (x &&!x)

Since \(x\ &&!x = F\) always, the expression simplifies to \((y\ && y)\ || F\). By the property of logical OR (\(a\ || F=a\)), when \(y = T\) (for the row \(x = F,y = T\)), \((y\ && y)\ || (x\ &&!x)=T\ || F=T\). When \(y = F\) (for the row \(x = F,y = F\)), \((y\ && y)\ || (x\ &&!x)=F\ || F=F\).

Answer:

When \(x = F,y = T\), the value is \(T\). When \(x = F,y = F\), the value is \(F\).