De Morgan’s laws
Use De Morgan’s laws to find the negation of the following statement.
Kwame will take a job in industry or go to graduate school.
| Kwame will not take a job in industry and will not go to graduate school. | |||
| Kwame will take a job in industry and will not go to graduate school. | |||
| Kwame will not take a job in industry and will go to graduate school. | |||
| Kwame will take a job in industry and will go to graduate school. | |||
| Kwame will not take a job in industry or will not go to graduate school. |
Complete the truth table for each of these compound propositions.
a) p→(¬q∧r)
| p | q | r | ¬q | ¬q∧r | p→(¬q∧r) |
| T | T | T | |||
| T | T | F | |||
| T | F | T | |||
| T | F | F | |||
| F | T | T | |||
| F | T | F | |||
| F | F | T | |||
| F | F | F |
b) ¬p→(q→r)
| p | q | r | ¬p | q→r | ¬p→(q→r) |
| T | T | T | |||
| T | T | F | |||
| T | F | T | |||
| T | F | F | |||
| F | T | T | |||
| F | T | F | |||
| F | F | T | |||
| F | F | F |
c) (p→q)∧(¬p→r)
| p | q | r | ¬p | p→q | ¬p→r | (p→q)∧(¬p→r) |
| T | T | T | ||||
| T | T | F | ||||
| T | F | T | ||||
| T | F | F | ||||
| F | T | T | ||||
| F | T | F | ||||
| F | F | T | ||||
| F | F | F |
d) (p↔q)∧(¬q↔r)
| p | q | r | ¬q | p↔q | ¬q↔r | (p↔q)∧(¬q↔r) |
| T | T | T | ||||
| T | T | F | ||||
| T | F | T | ||||
| T | F | F | ||||
| F | T | T | ||||
| F | T | F | ||||
| F | F | T | ||||
| F | F | F |
e) (¬p↔¬q)↔(q↔r)
| p | q | r | ¬p | ¬q | ¬p↔¬q | q↔r | (¬p↔¬q)↔(q↔r) |
| T | T | T | |||||
| T | T | F | |||||
| T | F | T | |||||
| T | F | F | |||||
| F | T | T | |||||
| F | T | F | |||||
| F | F | T | |||||
| F | F | F |
Show that each of these conditional statements is a tautology by completing the truth tables.
a) (p∧q)→p
| p | q | p∧q | (p∧q)→p |
| T | T | ||
| T | F | ||
| F | T | ||
| F | F |
b) q→(p∨q)
| p | q | p∨q | q→(p∨q) |
| T | T | ||
| T | F | ||
| F | T | ||
| F | F |
c) ¬p→(p→q)
| p | q | ¬p | p→q | ¬p→(p→q) |
| T | T | |||
| T | F | |||
| F | T | |||
| F | F |
d) (p∧q)→(p→q)
| p | q | p∧q | p→q | (p∧q)→(p→q) |
| T | T | |||
| T | F | |||
| F | T | |||
| F | F |
e) ¬(p→q)→p
| p | q | p→q | ¬(p→q) | ¬(p→q)→p |
| T | T | |||
| T | F | |||
| F | T | |||
| F | F |
f) ¬(p→q)→¬q
| p | q | p→q | ¬(p→q) | ¬q | ¬(p→q)→¬q |
| T | T | ||||
| T | F | ||||
| F | T | ||||
| F | F |
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