Question

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

Explanation:

Step1: Recall logical - operator rules

• $\land$ (conjunction): $A\land B$ is true only when both $A$ and $B$ are true.
• $\lor$ (disjunction): $A\lor B$ is true when either $A$ or $B$ or both are true.
• $

eg$ (negation): $
eg A$ has the opposite truth - value of $A$.

• $A

ightarrow B$ is false when $A$ is true and $B$ is false, and true otherwise.

Step2: Evaluate $(p\land r)

ightarrow(p\land q)$
Given $p = T$, $q = T$, $r = F$.

• First, $p\land r$: Since $r = F$, $p\land r=F$.
• Second, $p\land q$: Since $p = T$ and $q = T$, $p\land q=T$.
• Then, $F

ightarrow T$ is $T$.

Step3: Evaluate $(p\lor r)

ightarrow(p\lor q)$

• $p\lor r$: Since $p = T$, $p\lor r=T$.
• $p\lor q$: Since $p = T$ and $q = T$, $p\lor q=T$.
• $T

ightarrow T$ is $T$.

Step4: Evaluate $

eg p\land
eg r$

• $

eg p = F$ (because $p = T$), $
eg r = T$ (because $r = F$).

• $

eg p\land
eg r=F\land T = F$.

Step5: Evaluate $p

ightarrow r$
Since $p = T$ and $r = F$, $p
ightarrow r$ is $F$.

Step6: Evaluate $

eg(p\land r)$

• $p\land r=F$ (as $r = F$), so $

eg(p\land r)=T$.

Step7: Evaluate $r

ightarrow p$
Since $r = F$ and $p = T$, $r
ightarrow p$ is $T$.

Step8: Evaluate $p

ightarrow
eg(p\lor r)$

• $p\lor r=T$ (because $p = T$), $

eg(p\lor r)=F$.

• Since $p = T$ and $

eg(p\lor r)=F$, $p
ightarrow
eg(p\lor r)$ is $F$.

Step9: Evaluate $

eg p
ightarrow(p\lor r)$

• $

eg p = F$ (because $p = T$), $p\lor r=T$ (because $p = T$).

• $F

ightarrow T$ is $T$.

Answer:

• $(p\land r)

ightarrow(p\land q)$: TRUE

• $(p\lor r)

ightarrow(p\lor q)$: TRUE

• $

eg p\land
eg r$: FALSE

• $p

ightarrow r$: FALSE

• $

eg(p\land r)$: TRUE

• $r

ightarrow p$: TRUE

• $p

ightarrow
eg(p\lor r)$: FALSE

• $

eg p
ightarrow(p\lor r)$: TRUE