Geometry · Unit 1
Join propositions with AND, OR, and NOT, and say a symbolic sentence in words.
Proposition
A proposition is a sentence that is true or false. “A triangle has 3 sides” is a proposition. “Draw a triangle” is not.
Conjunction ∧ AND
p ∧ q is true only when both parts are true.
Disjunction ∨ OR
p ∨ q is true when at least one part is true.
Negation ∼
∼p is the opposite of p. Parentheses first, then negation, then ∧ and ∨.
Name the connective, then say the compound sentence in words.
p ∧ q means both parts are true.
Maya packed a sweater and Maya packed boots.
Answer: Maya packed a sweater and Maya packed boots.
p ∨ q means at least one part is true.
The light is on or the door is locked.
Answer: The light is on or the door is locked.
∼p is the opposite of p.
The polygon is not regular.
Answer: The polygon is not regular.
∧ is conjunction (AND).
∨ is disjunction (OR).
Answer: ∧
p ∧ q
Angle A is acute and angle B is obtuse.
Answer: Angle A is acute and angle B is obtuse.
∼p
AB is not parallel to CD.
Answer: AB is not parallel to CD.
Parentheses. Translate a compound proposition with three parts.
p: The trail is open.
q: It is raining.
r: We pack rain jackets.
(p ∨ q) ∧ r
The whole statement is true when r is true, and at least one of p or q is true.
The trail is open or it is raining, and we pack rain jackets.
Answer: The trail is open or it is raining, and we pack rain jackets.
p ∧ (q ∨ r)
Nia takes the bus, and Nia walks or Nia brings a lunch.
Answer: Nia takes the bus, and Nia walks or Nia brings a lunch.
∼(p ∧ q)
It is false that the figure is a square and a rhombus.
Answer: It is false that the figure is a square and a rhombus.
∼p ∨ q
It is not Friday, or the lab is due.
Answer: It is not Friday, or the lab is due.
The AND is outside, so r must be true.
At least one of p or q must be true.
Answer: r, and at least one of p or q
∼p ∧ ∼q
The kettle is not on and the tea is not ready.
Answer: The kettle is not on and the tea is not ready.
Match a sentence to symbols, or write a symbolic sentence in words.
p ∧ ∼q
The soup is hot and the bread is not ready.
Answer: The soup is hot and the bread is not ready.
“Or” joins oven and timer inside parentheses.
AND the bread is baking sits outside.
(p ∨ q) ∧ r
Answer: B
NOT raining is outside the OR.
Walk or bike stays in parentheses.
∼p ∧ (q ∨ r)
Answer: A
∼(p ∨ q)
It is false that the battery is charged or the tablet turns on.
That is the same as: the battery is not charged and the tablet does not turn on.
Answer: It is false that the battery is charged or the tablet turns on.
“Or” joins pencil and eraser inside parentheses.
AND paper is outside.
(p ∨ q) ∧ r
Answer: B
∼(p ∧ q)
It is false that the charger is plugged in and the phone is on.
Answer: It is false that the charger is plugged in and the phone is on.
(∼p ∨ q) ∧ r
The bus is not late or we walk, and we text.
Answer: The bus is not late or we walk, and we text.
∼p ∨ ∼q
The window is not open, or the fan is not on.
Answer: The window is not open, or the fan is not on.
p ∧ (q ∨ r)
The lock is set, and the alarm is on or we left.
Answer: The lock is set, and the alarm is on or we left.
∼(p ∨ q) ∧ r
It is false that Sam packed fruit or nuts, and Sam packed water.
Answer: It is false that Sam packed fruit or nuts, and Sam packed water.
∼p ∧ q
The printer does not have paper, and the printer has ink.
Answer: The printer does not have paper, and the printer has ink.
(p ∧ ∼q) ∨ r
The museum is open and it is not Monday, or we buy tickets.
Answer: The museum is open and it is not Monday, or we buy tickets.
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