P Q X 1 1 1 1 0 0 0 1 0 0 0 1
Truth Table Analysis: Finding Logical Operator X
Truth Table Breakdown
The provided truth table defines the output X based on inputs P and Q:
| P | Q | X |
| 1 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
Evaluating Logical Operators
We need to find the logical operator that matches the relationship between P, Q, and X.
1. Checking [P $\land$ Q] (AND)
- If P=1 and Q=1, P $\land$ Q = 1. (Matches X)
- If P=1 and Q=0, P $\land$ Q = 0. (Matches X)
- If P=0 and Q=1, P $\land$ Q = 0. (Matches X)
- If P=0 and Q=0, P $\land$ Q = 0. (Does NOT match X=1)
Therefore, [P $\land$ Q] is not the correct operator.
2. Checking [P $\lor$ Q] (OR)
- If P=1 and Q=0, P $\lor$ Q = 1. (Does NOT match X=0)
Therefore, [P $\lor$ Q] is not the correct operator.
3. Checking [P $\rightarrow$ Q] (Implication)
- If P=0 and Q=1, P $\rightarrow$ Q = 1. (Does NOT match X=0)
Therefore, [P $\rightarrow$ Q] is not the correct operator.
4. Checking [P $\leftrightarrow$ Q] (Biconditional / XNOR)
- If P=1 and Q=1, P $\leftrightarrow$ Q = 1. (Matches X)
- If P=1 and Q=0, P $\leftrightarrow$ Q = 0. (Matches X)
- If P=0 and Q=1, P $\leftrightarrow$ Q = 0. (Matches X)
- If P=0 and Q=0, P $\leftrightarrow$ Q = 1. (Matches X)
The biconditional operator [P $\leftrightarrow$ Q] perfectly matches the truth table output for X.
Conclusion
The output X in the truth table represents the biconditional logical operation between P and Q, denoted as [P $\leftrightarrow$ Q]. This operation is true only when P and Q have the same truth value (both true or both false).
Important Questions from Logical Deduction
- Based only on the conversation below, identify the logically correct inference:
“Even if I had known that you were in the hospital, I would not have gone there to see you", Ramya told Josephine. In the given figure, the numbers associated with the rectangle, triangle, and ellipse are 1, 2, and 3, respectively. Which one among the given options is the most appropriate combination of P, Q, and R ?
- Column-I has statements made by Shanthala; and, Column-II has responses given by Kanishk.
Column-I Column-II P. This house is in a mess. 1. Alright, I won't bring it up during our conversations. Q. I am not happy with the marks given to me. 2. Well, you can easily look it up. R. Politics is a subject I avoid talking about. 3. No problem, let me clear it up for you. S. I don't know what this word means. 4. Don't worry, I will take it up with your teacher.
Identify the option that has the correct match between Column-I and Column-II. A color model is shown in the figure with color codes: Yellow (Y), Magenta (M), Cyan (Cy), Red (R), Blue (Bl), Green (G), and Black (K).
Which one of the following options displays the color codes that are consistent with the color model?- Whenever, our pet Johny falls sick, he just sniffs the food but does not eat. If forced, he pisses on the side of the food-plate. Later, I came across a study conducted by SAN's School of Vets that such behaviour of pets is an indication of diabetes among dogs. This made me to conclude that Johny is suffering from diabetes.
Which one of the Mill's methods is used here for arriving at the conclusion?
