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 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

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 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

Accepted Answer

### Question 1
# Explanation:
## Step1: Define perfect squares
Perfect squares: $n^2$ where $n$ is integer.
## Step2: Find odd perfect square
Take $n=3$, $3^2=9$.
## Step3: Test divisibility by 2
$9$ is not divisible by 2.
# Answer:
False. Counterexample: $9$ (a perfect square, not divisible by 2)

---

### Question 2
# Explanation:
## Step1: Define multiples of 3
Multiples of 3: $3k$ where $k$ is integer.
## Step2: Find non-multiple of 6
Take $k=1$, $3(1)=3$.
## Step3: Test divisibility by 6
$3$ is not divisible by 6.
# Answer:
False. Counterexample: $3$ (a multiple of 3, not a multiple of 6)

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### Question 3
# Explanation:
## Step1: Define supplementary/congruent angles
Supplementary: sum to $180^\circ$; congruent: equal measure.
## Step2: Identify counterexample diagram
Diagram C has two $90^\circ$ angles: $90+90=180^\circ$ (supplementary) and equal (congruent).
# Answer:
C

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### Question 4
# Explanation:
## Step1: Translate statements to symbols
$p$: Memorial Day in July; $\sim q$: parallel lines intersect.
## Step2: Match "or" to logical symbol
"Or" uses $\lor$, so statement is $p \lor \sim q$.
# Answer:
D. $p \lor \sim q$

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### Question 5
# Explanation:
## Step1: Evaluate base statements
$p$ is false (Memorial Day is in May); $q$ is true (parallel lines never intersect).
## Step2: Test each compound statement
- "Memorial Day is in July or parallel lines intersect": $	ext{False} \lor 	ext{False} = 	ext{False}$
- "Memorial Day is not in July and parallel lines intersect": $	ext{True} \land 	ext{False} = 	ext{False}$
- "Memorial Day is in July or parallel lines never intersect": $	ext{False} \lor 	ext{True} = 	ext{True}$
- "Memorial Day is not in July or parallel lines intersect": $	ext{True} \lor 	ext{False} = 	ext{True}$
# Answer:
- $\square$ Memorial Day is in July or parallel lines intersect.
- $\square$ Memorial Day is not in July and parallel lines intersect.

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### Question 6
# Explanation:
## Step1: Calculate $p \lor \sim r$
For each row, $\sim r$ is opposite of $r$; $p \lor \sim r$ is true if either $p$ or $\sim r$ is true.
## Step2: Calculate $(p \lor \sim r) \land \sim q$
$\sim q$ is opposite of $q$; final value is true only if both $(p \lor \sim r)$ and $\sim q$ are true.
# Answer:
| $p$ | $q$ | $r$ | $\sim r$ | $p \lor \sim r$ | $\sim q$ | $(p \lor \sim r) \land \sim q$ |
|---|---|---|---|---|---|---|
| T | T | T | F | T | F | F |
| T | T | F | T | T | F | F |
| T | F | T | F | T | T | T |
| T | F | F | T | T | T | T |
| F | T | T | F | F | F | F |
| F | T | F | T | T | F | F |
| F | F | T | F | F | T | F |
| F | F | F | T | T | T | T |

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QUESTION IMAGE

Question

Explanation:

Question 1

Step1: Define perfect squares

Step2: Find odd perfect square

Step3: Test divisibility by 2

Step1: Define multiples of 3

Step2: Find non-multiple of 6

Step3: Test divisibility by 6

Step1: Define supplementary/congruent angles

Step2: Identify counterexample diagram

Answer:

Question 2