Question

determine if each symbolic statement is true or false when... p is true, q is true, and r is false
r → p
¬p → (p ∨ r)
¬p ∧ ¬r
p → ¬(p ∨ r)
p → r
¬(p ∧ r)
(p ∧ r) → (p ∧ q)
(p ∨ r) → (p ∨ q)

Explanation:

Step1: Recall logical - connective rules

• $

eg$ is negation (true becomes false and vice - versa), $\vee$ is disjunction (true if at least one is true), $\wedge$ is conjunction (true only if both are true), and $A
ightarrow B$ is false when $A$ is true and $B$ is false, otherwise true. Given $p = T$, $q = T$, $r = F$.

Step2: Evaluate $r

ightarrow p$

• For $r

ightarrow p$, since $r = F$, by the rule of implication, the statement is $T$.

Step3: Evaluate $

eg p
ightarrow(p\vee r)$

• $

eg p=F$ (because $p = T$), $p\vee r=T$ (since $p = T$), and $F
ightarrow T$ is $T$.

Step4: Evaluate $

eg p\wedge
eg r$

• $

eg p = F$, $
eg r=T$, and $F\wedge T$ is $F$.

Step5: Evaluate $p

ightarrow
eg(p\vee r)$

• $p = T$, $p\vee r=T$, $

eg(p\vee r)=F$, and $T
ightarrow F$ is $F$.

Step6: Evaluate $p

ightarrow r$

• $p = T$, $r = F$, so $p

ightarrow r$ is $F$.

Step7: Evaluate $

eg(p\wedge r)$

• $p\wedge r=F$ (because $r = F$), so $

eg(p\wedge r)=T$.

Step8: Evaluate $(p\wedge r)

ightarrow(p\wedge q)$

• $p\wedge r=F$, $p\wedge q=T$, and $F

ightarrow T$ is $T$.

Step9: Evaluate $(p\vee r)

ightarrow(p\vee q)$

• $p\vee r=T$, $p\vee q=T$, and $T

ightarrow T$ is $T$.

Answer:

$r
ightarrow p$: $T$
$
eg p
ightarrow(p\vee r)$: $T$
$
eg p\wedge
eg r$: $F$
$p
ightarrow
eg(p\vee r)$: $F$
$p
ightarrow r$: $F$
$
eg(p\wedge r)$: $T$
$(p\wedge r)
ightarrow(p\wedge q)$: $T$
$(p\vee r)
ightarrow(p\vee q)$: $T$