Question

directions: use the statements below along with the diagram to write compound statements. then give its truth value.
p: points c, e, and b are collinear.
q: ∠aec≅∠deb
r: ef is the angle bisector of ∠aed
s: ∠bec is an acute angle.

1. p ∨ q: points c, e, and b are collinear or ∠aec≅∠deb

truth value: t

1. q ∧ s:

truth value:

1. ~p ∧ r:

truth value:

1. r ∨ ~s:

truth value:

1. ~q ∧ ~r:

truth value:

1. p ∨ ~q:

truth value:

1. ~r ∨ ~s:

truth value:

1. ~q ∧ s:

truth value:

1. p ∨ r:

truth value:

Explanation:

Step1: Recall logical - connective meanings

The symbol $\vee$ means "or" (disjunction), $\wedge$ means "and" (conjunction), and $\sim$ means "not" (negation).

Step2: Analyze $q\wedge s$

The compound statement $q\wedge s$ is " $\angle AEC\cong\angle DEB$ and $\angle BEC$ is an acute - angle". Assume from the diagram that $\angle AEC$ and $\angle DEB$ are vertical angles (so $q$ is true) and $\angle BEC$ is acute (so $s$ is true). Since both $q$ and $s$ are true, by the definition of conjunction, $q\wedge s$ is true.

Step3: Analyze $\sim p\wedge r$

The statement $\sim p$ is "Points $C$, $E$, and $B$ are not collinear" and $r$ is " $\overline{EF}$ is the angle - bisector of $\angle AED$". If $p$ is true (points $C$, $E$, $B$ are collinear), then $\sim p$ is false. For a conjunction $\sim p\wedge r$ to be true, both $\sim p$ and $r$ must be true. Since $\sim p$ is false, $\sim p\wedge r$ is false.

Step4: Analyze $r\vee\sim s$

The statement $\sim s$ is " $\angle BEC$ is not an acute - angle". If $r$ is true ( $\overline{EF}$ is the angle - bisector of $\angle AED$) or $\sim s$ is true, then $r\vee\sim s$ is true.

Step5: Analyze $\sim q\wedge\sim r$

The statement $\sim q$ is " $\angle AEC
ot\cong\angle DEB$" and $\sim r$ is " $\overline{EF}$ is not the angle - bisector of $\angle AED$". For $\sim q\wedge\sim r$ to be true, both $\sim q$ and $\sim r$ must be true.

Step6: Analyze $p\vee\sim q$

The statement $\sim q$ is " $\angle AEC
ot\cong\angle DEB$". If $p$ is true (points $C$, $E$, $B$ are collinear) or $\sim q$ is true, then $p\vee\sim q$ is true.

Step7: Analyze $\sim r\vee\sim s$

The statement $\sim r$ is " $\overline{EF}$ is not the angle - bisector of $\angle AED$" and $\sim s$ is " $\angle BEC$ is not an acute - angle". If either $\sim r$ or $\sim s$ is true, then $\sim r\vee\sim s$ is true.

Step8: Analyze $\sim q\wedge s$

The statement $\sim q$ is " $\angle AEC
ot\cong\angle DEB$" and $s$ is " $\angle BEC$ is an acute - angle". For $\sim q\wedge s$ to be true, both $\sim q$ and $s$ must be true.

Step9: Analyze $p\vee r$

The statement $p$ is "Points $C$, $E$, and $B$ are collinear" and $r$ is " $\overline{EF}$ is the angle - bisector of $\angle AED$". If either $p$ or $r$ is true, then $p\vee r$ is true.

1. Compound Statement: $\angle AEC\cong\angle DEB$ and $\angle BEC$ is an acute - angle. Truth Value: T
2. Compound Statement: Points $C$, $E$, and $B$ are not collinear and $\overline{EF}$ is the angle - bisector of $\angle AED$. Truth Value: F
3. Compound Statement: $\overline{EF}$ is the angle - bisector of $\angle AED$ or $\angle BEC$ is not an acute - angle. Truth Value: Depends on the diagram (assume T if $r$ is true)
4. Compound Statement: $\angle AEC

ot\cong\angle DEB$ and $\overline{EF}$ is not the angle - bisector of $\angle AED$. **Truth Value**: Depends on the diagram (assume F if $q$ and $r$ are true in the diagram)

1. Compound Statement: Points $C$, $E$, and $B$ are collinear or $\angle AEC

ot\cong\angle DEB$. **Truth Value**: T (assuming $p$ is true)

1. Compound Statement: $\overline{EF}$ is not the angle - bisector of $\angle AED$ or $\angle BEC$ is not an acute - angle. Truth Value: Depends on the diagram
2. Compound Statement: $\angle AEC

ot\cong\angle DEB$ and $\angle BEC$ is an acute - angle. **Truth Value**: Depends on the diagram (assume F if $q$ is true)

1. Compound Statement: Points $C$, $E$, and $B$ are collinear or $\overline{EF}$ is the angle - bisector of $\angle AED$. Truth Value: T (assuming $p$ is true)

Answer:

Step1: Recall logical - connective meanings

The symbol $\vee$ means "or" (disjunction), $\wedge$ means "and" (conjunction), and $\sim$ means "not" (negation).

Step2: Analyze $q\wedge s$

The compound statement $q\wedge s$ is " $\angle AEC\cong\angle DEB$ and $\angle BEC$ is an acute - angle". Assume from the diagram that $\angle AEC$ and $\angle DEB$ are vertical angles (so $q$ is true) and $\angle BEC$ is acute (so $s$ is true). Since both $q$ and $s$ are true, by the definition of conjunction, $q\wedge s$ is true.

Step3: Analyze $\sim p\wedge r$

The statement $\sim p$ is "Points $C$, $E$, and $B$ are not collinear" and $r$ is " $\overline{EF}$ is the angle - bisector of $\angle AED$". If $p$ is true (points $C$, $E$, $B$ are collinear), then $\sim p$ is false. For a conjunction $\sim p\wedge r$ to be true, both $\sim p$ and $r$ must be true. Since $\sim p$ is false, $\sim p\wedge r$ is false.

Step4: Analyze $r\vee\sim s$

The statement $\sim s$ is " $\angle BEC$ is not an acute - angle". If $r$ is true ( $\overline{EF}$ is the angle - bisector of $\angle AED$) or $\sim s$ is true, then $r\vee\sim s$ is true.

Step5: Analyze $\sim q\wedge\sim r$

The statement $\sim q$ is " $\angle AEC
ot\cong\angle DEB$" and $\sim r$ is " $\overline{EF}$ is not the angle - bisector of $\angle AED$". For $\sim q\wedge\sim r$ to be true, both $\sim q$ and $\sim r$ must be true.

Step6: Analyze $p\vee\sim q$

The statement $\sim q$ is " $\angle AEC
ot\cong\angle DEB$". If $p$ is true (points $C$, $E$, $B$ are collinear) or $\sim q$ is true, then $p\vee\sim q$ is true.

Step7: Analyze $\sim r\vee\sim s$

The statement $\sim r$ is " $\overline{EF}$ is not the angle - bisector of $\angle AED$" and $\sim s$ is " $\angle BEC$ is not an acute - angle". If either $\sim r$ or $\sim s$ is true, then $\sim r\vee\sim s$ is true.

Step8: Analyze $\sim q\wedge s$

The statement $\sim q$ is " $\angle AEC
ot\cong\angle DEB$" and $s$ is " $\angle BEC$ is an acute - angle". For $\sim q\wedge s$ to be true, both $\sim q$ and $s$ must be true.

Step9: Analyze $p\vee r$

The statement $p$ is "Points $C$, $E$, and $B$ are collinear" and $r$ is " $\overline{EF}$ is the angle - bisector of $\angle AED$". If either $p$ or $r$ is true, then $p\vee r$ is true.

1. Compound Statement: $\angle AEC\cong\angle DEB$ and $\angle BEC$ is an acute - angle. Truth Value: T
2. Compound Statement: Points $C$, $E$, and $B$ are not collinear and $\overline{EF}$ is the angle - bisector of $\angle AED$. Truth Value: F
3. Compound Statement: $\overline{EF}$ is the angle - bisector of $\angle AED$ or $\angle BEC$ is not an acute - angle. Truth Value: Depends on the diagram (assume T if $r$ is true)
4. Compound Statement: $\angle AEC

ot\cong\angle DEB$ and $\overline{EF}$ is not the angle - bisector of $\angle AED$. **Truth Value**: Depends on the diagram (assume F if $q$ and $r$ are true in the diagram)

1. Compound Statement: Points $C$, $E$, and $B$ are collinear or $\angle AEC

ot\cong\angle DEB$. **Truth Value**: T (assuming $p$ is true)

1. Compound Statement: $\overline{EF}$ is not the angle - bisector of $\angle AED$ or $\angle BEC$ is not an acute - angle. Truth Value: Depends on the diagram
2. Compound Statement: $\angle AEC

ot\cong\angle DEB$ and $\angle BEC$ is an acute - angle. **Truth Value**: Depends on the diagram (assume F if $q$ is true)

1. Compound Statement: Points $C$, $E$, and $B$ are collinear or $\overline{EF}$ is the angle - bisector of $\angle AED$. Truth Value: T (assuming $p$ is true)