Answer to Question #327315 in Discrete Mathematics for Dreaper

Question

Show your solution.

1.     Show, by the use of the truth table/matrix, that the statement (p∨q)∨ (¬q) is tautology.

2.     Show that p ↔ q and (p ∧ q) ∨ (¬p ∧ ¬q) are logically equivalent.

Answer 1

1.

$p~~~~~~q~~~~~~p\lor q~~~~~~\lnot q~~~~~~(p\lor q)\lor(\lnot q)\\ 0~~~~~~0~~~~~~~~~0~~~~~~~~~~~1~~~~~~~~~~~~~~~~~~1\\ 0~~~~~~1~~~~~~~~~1~~~~~~~~~~~0~~~~~~~~~~~~~~~~~~1\\ 1~~~~~~0~~~~~~~~~1~~~~~~~~~~~1~~~~~~~~~~~~~~~~~~1\\ 1~~~~~~1~~~~~~~~~1~~~~~~~~~~~0~~~~~~~~~~~~~~~~~~1$

We can see that $(p\lor q)\lor(\lnot q)$ is always true meaning that it is a tautology.

2.

$p~~~~~~q~~~~~~p\land q~~~~~~\lnot p\land\lnot q~~~~~~(p\land q)\lor(\lnot p\land\lnot q)\\ 0~~~~~~0~~~~~~~~~0~~~~~~~~~~~~~~~1~~~~~~~~~~~~~~~~~~~~~~~~~~1\\ 0~~~~~~1~~~~~~~~~0~~~~~~~~~~~~~~~0~~~~~~~~~~~~~~~~~~~~~~~~~~0\\ 1~~~~~~0~~~~~~~~~0~~~~~~~~~~~~~~~0~~~~~~~~~~~~~~~~~~~~~~~~~~0\\ 1~~~~~~1~~~~~~~~~1~~~~~~~~~~~~~~~0~~~~~~~~~~~~~~~~~~~~~~~~~~1$

$p~~~~~~q~~~~~~p\lrarr q\\ 0~~~~~~0~~~~~~~~~~1\\ 0~~~~~~1~~~~~~~~~~0\\ 1~~~~~~0~~~~~~~~~~0\\ 1~~~~~~1~~~~~~~~~~1$

Comparing two last tables we can see that $(p\land q)\lor(\lnot p\land\lnot q)=p\lrarr q$