Question

name: tyler smithdate:_______ per:_____unit 2 testlogic and prooffor questions 1-2, determine if the conjectures are true or false. if false, provide a counterexample.1. all perfect squares are divisible by 2.2. multiples of 3 are always multiples of 6.3. which diagram provides a counterexample to the statement below?“supplementary angles are never congruent.”a. b. c. d.use the statements below to answer questions 4 and 5.p: memorial day is in july. q: parallel lines never intersect.4. which represents the symbolic notation of the compound statement below?“memorial day is in july or parallel lines intersect.”a. $p \land q$b. $p \land \sim q$c. $p \lor q$d. $p \lor \sim q$5. check the statements below that are false.$square$ memorial day is in july or parallel lines intersect.$square$ memorial day is not in july and parallel lines never intersect.$square$ memorial day is not in july and parallel lines intersect.$square$ memorial day is in july or parallel lines never intersect.$square$ memorial day is not in july or parallel lines never intersect.6. complete a truth table given the compound statement below.given: $(p \lor \sim r) \land \sim q$| $p$ | $q$ | $r$ ||-----|-----|-----|| t | t | t || t | t | f || t | f | t || t | f | f || f | t | t || f | t | f || f | f | t || f | f | f |© gina wilson (all things algebra®, llc), 2014-2020

Explanation:

Step1: Evaluate conjecture 1

A perfect square like $9=3^2$ is not divisible by 2.

Step2: Evaluate conjecture 2

A multiple of 3 like 9 is not a multiple of 6.

Step3: Identify counterexample for Q3

Congruent supplementary angles are both $90^\circ$ (diagram C shows this).

Step4: Translate statement to logic (Q4)

"Memorial Day is in July" = $p$, "parallel lines intersect" = $\sim q$. The "or" is $\lor$, so $p \lor \sim q$.

Step5: Evaluate false statements (Q5)

$p$ is false (Memorial Day is in May), $q$ is true.

• "Memorial Day is in July or parallel lines never intersect": $p \lor q$ = $\text{False} \lor \text{True} = \text{True}$
• "Memorial Day is not in July and parallel lines intersect": $\sim p \land \sim q$ = $\text{True} \land \text{False} = \text{False}$
• "Memorial Day is in July or parallel lines intersect": $p \lor \sim q$ = $\text{False} \lor \text{False} = \text{False}$
• "Memorial Day is not in July and parallel lines never intersect": $\sim p \land q$ = $\text{True} \land \text{True} = \text{True}$
• "Memorial Day is in July or parallel lines never intersect": $p \lor q$ = $\text{True}$ (already noted)

Step6: Build truth table (Q6)

Calculate $\sim q$, $p \lor \sim r$, then $(p \lor \sim r) \land \sim q$ row by row.

1. Row 1 ($T,T,T$): $\sim q=F$, $\sim r=F$, $p \lor \sim r=T\lor F=T$, $(p \lor \sim r) \land \sim q=T\land F=F$
2. Row 2 ($T,T,F$): $\sim q=F$, $\sim r=T$, $p \lor \sim r=T\lor T=T$, $(p \lor \sim r) \land \sim q=T\land F=F$
3. Row 3 ($T,F,T$): $\sim q=T$, $\sim r=F$, $p \lor \sim r=T\lor F=T$, $(p \lor \sim r) \land \sim q=T\land T=T$
4. Row 4 ($T,F,F$): $\sim q=T$, $\sim r=T$, $p \lor \sim r=T\lor T=T$, $(p \lor \sim r) \land \sim q=T\land T=T$
5. Row 5 ($F,T,T$): $\sim q=F$, $\sim r=F$, $p \lor \sim r=F\lor F=F$, $(p \lor \sim r) \land \sim q=F\land F=F$
6. Row 6 ($F,T,F$): $\sim q=F$, $\sim r=T$, $p \lor \sim r=F\lor T=T$, $(p \lor \sim r) \land \sim q=T\land F=F$
7. Row 7 ($F,F,T$): $\sim q=T$, $\sim r=F$, $p \lor \sim r=F\lor F=F$, $(p \lor \sim r) \land \sim q=F\land T=F$
8. Row 8 ($F,F,F$): $\sim q=T$, $\sim r=T$, $p \lor \sim r=F\lor T=T$, $(p \lor \sim r) \land \sim q=T\land T=T$

Answer:

1. False. Counterexample: $9$ (a perfect square, not divisible by 2)
2. False. Counterexample: $9$ (a multiple of 3, not 6)
3. C
4. D. $p \lor \sim q$

5.

• Memorial Day is in July or parallel lines intersect.
• Memorial Day is not in July and parallel lines intersect.

1. Completed truth table:

$p$	$q$	$r$	$\sim q$	$\sim r$	$p \lor \sim r$	$(p \lor \sim r) \land \sim q$
T	T	F	F	T	T	F
T	F	T	T	F	T	T
T	F	F	T	T	T	T
F	T	T	F	F	F	F
F	T	F	F	T	T	F
F	F	T	T	F	F	F
F	F	F	T	T	T	T