"p implies q" is a logical statement that means if p is true, then q must also be true. In other words, p is a sufficient condition for q to be true. This statement is also known as a conditional statement or an implication.
The statements "p implies q" and "q implies r" are related through a logical connective called "or". This means that at least one of the statements must be true for the overall statement to be true. In this case, either p being true implies q being true, or q being true implies r being true.
Yes, it is possible for both "p implies q" and "q implies r" to be false. This would happen if p is true but q is false, or if q is true but r is false. In this case, the overall statement would also be false because at least one of the individual statements is false.
"p implies q" and "q implies p" are not the same because they have different truth values. "p implies q" is true when p is true and q is true, whereas "q implies p" is true when q is true and p is true. The order of the statements matters in conditional statements.
The truth table for "p implies q" or "q implies r" would have four rows, representing all possible combinations of truth values for p, q, and r. The overall statement would be true if at least one of the individual statements is true, and it would be false only if both individual statements are false. The truth table would look like this:
| p | q | r | (p implies q) or (q implies r) |
|---|---|---|---|
| true | true | true | true |
| true | true | false | true |
| true | false | true | true |
| true | false | false | false |