Question

unit 2: logic & proof
homework 2: compound statements
this is a 2 - page document!
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: (overrightarrow{ef}) is the angle bisector of ∠aed
s: ∠bec is an acute angle.

1. (pvee q):

truth value:

1. (qwedge s):

truth value:

1. (sim pwedge r):

truth value:

1. (rveesim s):

truth value:

1. (sim qwedgesim r):

truth value:

1. (pveesim q):

truth value:

1. (sim rveesim s):

truth value:

1. (sim qwedge s):

truth value:

1. (pvee r):

truth value:

Explanation:

Step1: Analyze the truth values of individual statements

• \(p\) (Points \(C\), \(E\), and \(B\) are collinear) is False (\(F\)).
• \(q\) (\(\angle AEC\cong\angle DEB\)) is True (\(T\)) (vertical angles are congruent).
• \(r\) (\(\overrightarrow{EF}\) is the angle - bisector of \(\angle AED\)) is True (\(T\)).
• \(s\) (\(\angle BEC\) is an acute angle) is False (\(F\)).

Step2: Use logical connective rules

• For \(p\vee q\): The disjunction \(A\vee B\) is True if either \(A\) or \(B\) is True. Since \(q = T\), \(p\vee q\) is True.
• For \(q\wedge s\): The conjunction \(A\wedge B\) is True only if both \(A\) and \(B\) are True. Since \(s = F\), \(q\wedge s\) is False.
• For \(\sim p\wedge r\): \(\sim p=T\) (because \(p = F\)), and \(r = T\). The conjunction of two True statements is True.
• For \(r\vee\sim s\): \(\sim s=T\) (because \(s = F\)), and \(r = T\). The disjunction of two True statements is True.
• For \(\sim q\wedge\sim r\): \(\sim q=F\) (because \(q = T\)), so the conjunction (since one of the conjuncts is False) is False.
• For \(p\vee\sim q\): \(\sim q=F\) (because \(q = T\)), and \(p = F\). The disjunction of two False statements is False.
• For \(\sim r\vee\sim s\): \(\sim r=F\) (because \(r = T\)), \(\sim s=T\) (because \(s = F\)). The disjunction of a False and a True statement is True.
• For \(\sim q\wedge s\): \(\sim q=F\) (because \(q = T\)), so the conjunction is False.
• For \(p\vee r\): \(p = F\), \(r = T\). The disjunction of a False and a True statement is True.

Answer:

1. \(p\vee q\): Points \(C\), \(E\), and \(B\) are collinear or \(\angle AEC\cong\angle DEB\); Truth Value: \(T\)
2. \(q\wedge s\): \(\angle AEC\cong\angle DEB\) and \(\angle BEC\) is an acute angle; Truth Value: \(F\)
3. \(\sim p\wedge r\): Points \(C\), \(E\), and \(B\) are not collinear and \(\overrightarrow{EF}\) is the angle bisector of \(\angle AED\); Truth Value: \(T\)
4. \(r\vee\sim s\): \(\overrightarrow{EF}\) is the angle bisector of \(\angle AED\) or \(\angle BEC\) is not an acute angle; Truth Value: \(T\)
5. \(\sim q\wedge\sim r\): \(\angle AEC

ot\cong\angle DEB\) and \(\overrightarrow{EF}\) is not the angle bisector of \(\angle AED\); Truth Value: \(F\)

1. \(p\vee\sim q\): Points \(C\), \(E\), and \(B\) are collinear or \(\angle AEC

ot\cong\angle DEB\); Truth Value: \(F\)

1. \(\sim r\vee\sim s\): \(\overrightarrow{EF}\) is not the angle bisector of \(\angle AED\) or \(\angle BEC\) is not an acute angle; Truth Value: \(T\)
2. \(\sim q\wedge s\): \(\angle AEC

ot\cong\angle DEB\) and \(\angle BEC\) is an acute angle; Truth Value: \(F\)

1. \(p\vee r\): Points \(C\), \(E\), and \(B\) are collinear or \(\overrightarrow{EF}\) is the angle bisector of \(\angle AED\); Truth Value: \(T\)