Discrete mathematics
Find the truth table of Pv~(p^q)
To find the truth table of Pv~(p^q), we need to consider all possible combinations of truth values for p and q, and evaluate the expression for each combination.
Let's build the truth table step by step:
p | q | ~(p^q) | Pv~(p^q)
-----------------------
T | T | F | T
T | F | T | T
F | T | T | F
F | F | T | T
In the table above, we consider all possible combinations of truth values for p and q. For each combination, we evaluate ~(p^q) and Pv~(p^q).
Here's a step-by-step explanation of how we arrive at the truth table:
1. p = T, q = T:
- p^q = T^T = T
- ~(p^q) = ~T = F
- Pv~(p^q) = T v F = T
2. p = T, q = F:
- p^q = T^F = F
- ~(p^q) = ~F = T
- Pv~(p^q) = T v T = T
3. p = F, q = T:
- p^q = F^T = F
- ~(p^q) = ~F = T
- Pv~(p^q) = F v T = F
4. p = F, q = F:
- p^q = F^F = F
- ~(p^q) = ~F = T
- Pv~(p^q) = F v T = T
Thus, the truth table of Pv~(p^q) is:
p | q | ~(p^q) | Pv~(p^q)
-----------------------
T | T | F | T
T | F | T | T
F | T | T | F
F | F | T | T
Discrete mathematics
Verify that the proposition pv~(p^q) is a tantology
To verify that the proposition Pv~(p^q) is a tautology, we need to show that it is always true, regardless of the truth values of p and q.
From the truth table we constructed earlier, we can see that for every combination of truth values for p and q, the expression Pv~(p^q) evaluates to true (T). Therefore, the proposition is indeed a tautology.
In other words, no matter what values we assign to p and q, the proposition will always be true.