How do you fill in the blank on the following truth tables, true or false ?
| P | Q | P \land Q |
| T | F | ? |
P \land Q states P \text{ AND }Q.
This is true if and only if both P and Q are true.
However Q is false. Therefore P \land Q is false.
The correct response is:
| P | Q | P \land Q |
| T | F | F |
| P | Q | P\lor Q |
| F | T | ? |
P\lor Q states P OR Q.
Therefore P\lor Q is true if either P or Q is true.
Q is true, therefore P\lor Q is true.
The correct response is:
| P | Q | P\lor Q |
| F | T | T |
| P | Q | P\land Q |
| ? | T | T |
P\land Q states P and Q. It is true if and only if both P and Q are true.
Therefore P must be true.
The correct response is:
| P | Q | P \land Q |
| T | T | T |
| P | Q | P\lor Q |
| F | ? | F |
P\lor Q states P or Q.
It is false if and only if both P and Q are false.
Therefore Q is false.
The correct response is:
| P | Q | P\lor Q |
| F | F | F |
| P | Q | \left(P \lor Q\right) \land Q |
| T | F | ? |
P\lor Q states P or Q. Since P is true, we conclude P\lor Q is true.
\left(P\lor Q\right) \land Q states (P or Q) and Q. Both P or Q, and Q must be true for \left(P\lor Q\right) \land Q to be true. However Q is false, therefore \left(P\lor Q\right) \land Q is also false.
The correct response is:
| P | Q | \left( P \lor Q \right) \land Q |
| T | F | F |
| P | Q | \left(P \lor Q\right) \lor Q |
| ? | F | T |
\left(P \lor Q\right) \lor Q states (P or Q ) or Q.
The truth value is true, therefor either (P or Q) is true or Q is true.
However we are given that Q is false. Therefore (P or Q) must be true.
Since Q is false P must be true.
The correct response is:
| P | Q | \left(P \lor Q\right) \lor Q |
| T | F | T |
| P | Q | \left( P \land Q \right) \lor P |
| T | F | ? |
\left(P \land Q\right) \lor P states (P and Q) or P. Q is false, therefore P\land Q is false.
Additionally P is true, meaning \left(P \land Q\right) \lor P must be true.
The correct response is:
| P | Q | \left(P \land Q\right) \lor P |
| T | F | T |