Solved Examples on Truth Values and Truth Tables

Samuel Dominic Chukwuemeka (SamDom For Peace) (1.) For each logical proposition, state the truth value, giving reason(s).

(2.) For each compound logical statement:
(a.) Identify each logical statement and the logical connective.
(b.) Give reason(s) for your answer.

(3.) Draw truth tables for the compound propositions written in symbolic logic.
Show all work.

(1.) Politician: If elected, I promise to protect Medicare and expand education funding.
Politician is elected. He expands education funding but does not protect Medicare.


This is a compound logical statement.
The logical statements and the connective is:
p: Protect Medicare.
q: Expand educational funding.
Connective: and (conjunction)

He expands education funding but does not protect Medicare.
p is False and q is True.
The compound logical statement is: False because for the conjunction connective, if any of the statements is False, the compound statement is False.
(2.)

(3.)


(4.)


(5.)


(6.)


(11.) $p \lor \neg q$


$ p \lor \neg q \\[3ex] Main\:\: connective:\:\: \lor \\[3ex] First:\:\: p \\[3ex] Second:\:\: \neg q \\[3ex] $
$p$ $q$ $\neg q$ $p \lor \neg q$
$T$ $T$ $F$ $T$
$T$ $F$ $T$ $T$
$F$ $T$ $F$ $F$
$F$ $F$ $T$ $T$

(12.) $\neg p \lor \neg q$


$ \neg p \lor \neg q \\[3ex] Main\:\: connective:\:\: \lor \\[3ex] First:\:\: \neg p \\[3ex] Second:\:\: \neg q \\[3ex] $
$p$ $q$ $\neg p$ $\neg q$ $\neg p \lor \neg q$
$T$ $T$ $F$ $F$ $F$
$T$ $F$ $F$ $T$ $T$
$F$ $T$ $T$ $F$ $T$
$F$ $F$ $T$ $T$ $T$

(13.) $\neg(p \land \neg q)$


$ \neg(p \land \neg q) \\[3ex] Main\:\: connective:\:\: \neg \\[3ex] Other\:\: connective:\:\: \land \\[3ex] First:\:\: p \\[3ex] Second:\:\: \neg q \\[3ex] $
$p$ $q$ $\neg q$ $p \land \neg q$ $\neg(p \land \neg q)$
$T$ $T$ $F$ $F$ $T$
$T$ $F$ $T$ $T$ $F$
$F$ $T$ $F$ $F$ $T$
$F$ $F$ $T$ $F$ $T$

(14.) $p \rightarrow \neg p$


$ p \rightarrow \neg p \\[3ex] Main\:\: connective:\:\: \rightarrow \\[3ex] First:\:\: p \\[3ex] Second:\:\: \neg p \\[3ex] $
$p$ $\neg p$ $p \rightarrow \neg p$
$T$ $F$ $F$
$F$ $T$ $T$

(15.) $p \leftrightarrow \neg p$


$ p \leftrightarrow \neg p \\[3ex] Main\:\: connective:\:\: \leftrightarrow \\[3ex] First:\:\: p \\[3ex] Second:\:\: \neg p \\[3ex] $
$p$ $\neg p$ $p \leftrightarrow \neg p$
$T$ $F$ $F$
$F$ $T$ $F$

(16.) $p \rightarrow \neg q$


$ p \rightarrow \neg q \\[3ex] Main\:\: connective:\:\: \rightarrow \\[3ex] First:\:\: p \\[3ex] Second:\:\: \neg q \\[3ex] $
$p$ $q$ $\neg q$ $p \rightarrow \neg q$
$T$ $T$ $F$ $F$
$T$ $F$ $T$ $T$
$F$ $T$ $F$ $T$
$F$ $F$ $T$ $T$

(17.) $p \leftrightarrow \neg q$


$ p \leftrightarrow \neg q \\[3ex] Main\:\: connective:\:\: \leftrightarrow \\[3ex] First:\:\: p \\[3ex] Second:\:\: \neg q \\[3ex] $
$p$ $q$ $\neg q$ $p \leftrightarrow \neg q$
$T$ $T$ $F$ $F$
$T$ $F$ $T$ $T$
$F$ $T$ $F$ $T$
$F$ $F$ $T$ $F$

(18.) $(p \rightarrow q) \rightarrow (q \rightarrow p)$


$ (p \rightarrow q) \rightarrow (q \rightarrow p) \\[3ex] Main\:\: connective:\:\: \rightarrow \\[3ex] First:\:\: p \rightarrow q \\[3ex] Second:\:\: q \rightarrow p \\[3ex] $
$p$ $q$ $p \rightarrow q$ $q \rightarrow p$ $(p \rightarrow q) \rightarrow (q \rightarrow p)$
$T$ $T$ $T$ $T$ $T$
$T$ $F$ $F$ $T$ $T$
$F$ $T$ $T$ $F$ $F$
$F$ $F$ $T$ $T$ $T$

(19.) $\neg q \lor (p \land r)$


$ \neg q \lor (p \land r) \\[3ex] Main\:\: connective:\:\: \lor \\[3ex] First:\:\: \neg q \\[3ex] Second:\:\: p \\[3ex] Third:\:\: r \\[3ex] $
$p$ $q$ $r$ $\neg q$ $p \land r$ $\neg q \lor (p \land r)$
$T$ $T$ $T$ $F$ $T$ $T$
$T$ $T$ $F$ $F$ $F$ $F$
$T$ $F$ $T$ $T$ $T$ $T$
$T$ $F$ $F$ $T$ $F$ $T$
$F$ $T$ $T$ $F$ $F$ $F$
$F$ $T$ $F$ $F$ $F$ $F$
$F$ $F$ $T$ $T$ $F$ $T$
$F$ $F$ $F$ $T$ $F$ $T$

(20.) $(r \lor \neg p) \land \neg q$


$ (r \lor \neg p) \land \neg q \\[3ex] Main\:\: connective:\:\: \land \\[3ex] First:\:\: r \\[3ex] Second:\:\: \neg p \\[3ex] Third:\:\: \neg q \\[3ex] $
$p$ $q$ $r$ $\neg p$ $r \lor \neg p$ $\neg q$ $(r \lor \neg p) \land \neg q$
$T$ $T$ $T$ $F$ $T$ $F$ $F$
$T$ $T$ $F$ $F$ $F$ $F$ $F$
$T$ $F$ $T$ $F$ $T$ $T$ $T$
$T$ $F$ $F$ $F$ $F$ $T$ $F$
$F$ $T$ $T$ $T$ $T$ $F$ $F$
$F$ $T$ $F$ $T$ $T$ $F$ $F$
$F$ $F$ $T$ $T$ $T$ $T$ $T$
$F$ $F$ $F$ $T$ $T$ $T$ $T$


(21.) $[(q \land \neg r) \land (\neg p \lor \neg q)] \lor (p \lor \neg r)$


$ [(q \land \neg r) \land (\neg p \lor \neg q)] \lor (p \lor \neg r) \\[3ex] Main\:\: connective:\:\: \lor \\[3ex] First:\:\: q \land \neg r \\[3ex] Second:\:\: \neg p \lor \neg q \\[3ex] Third:\:\: p \lor \neg r \\[3ex] $
$p$ $q$ $r$ $\neg p$ $\neg q$ $\neg r$ $(q \land \neg r)$ $(\neg p \lor \neg q)$ $(p \lor \neg r)$ $[(q \land \neg r) \land (\neg p \lor \neg q)]$ $[(q \land \neg r) \land (\neg p \lor \neg q)] \lor (p \lor \neg r)$
$T$ $T$ $T$ $F$ $F$ $F$ $F$ $F$ $T$ $F$ $T$
$T$ $T$ $F$ $F$ $F$ $T$ $T$ $F$ $T$ $F$ $T$
$T$ $F$ $T$ $F$ $T$ $F$ $F$ $T$ $T$ $F$ $T$
$T$ $F$ $F$ $F$ $T$ $T$ $F$ $T$ $T$ $F$ $T$
$F$ $T$ $T$ $T$ $F$ $F$ $F$ $T$ $F$ $F$ $F$
$F$ $T$ $F$ $T$ $F$ $T$ $T$ $T$ $T$ $T$ $T$
$F$ $F$ $T$ $T$ $T$ $F$ $F$ $T$ $F$ $F$ $F$
$F$ $F$ $F$ $T$ $T$ $T$ $F$ $T$ $T$ $F$ $T$