Flashcards · 13 cards

Logic

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Rule
What is ${\sim}p$ ?
Answer
The negation, "not $p$ ": true exactly when $p$ is false.
p ∼p
T F
F T
Negation ∼pThe opposite of p
From the lesson: Statements and implications
Rule
When is $p \wedge q$ true?
Answer
Only when both $p$ and $q$ are true.
p q p ∧ q
T T T
T F F
F T F
F F F
And: p ∧ qTrue only if both are true
From the lesson: Statements and implications
Rule
When is $p \vee q$ true?
Answer
When at least one is true, including when both are.
p q p ∨ q
T T T
T F T
F T T
F F F
Or: p ∨ qTrue if at least one is true
From the lesson: Statements and implications
Rule
When is $p \Rightarrow q$ false?
Answer
Only when $p$ is true and $q$ is false.
p q p ⇒ q
T T T
T F F
F T T
F F T
If … then: p ⇒ qFalse only for T ⇒ F
From the lesson: Statements and implications
Rule
When is $p \Leftrightarrow q$ true?
Answer
When $p$ and $q$ match: both true or both false.
p q p ⇔ q
T T T
T F F
F T F
F F T
If and only if: p ⇔ qTrue when they match
From the lesson: Statements and implications
Know it
For $p \Rightarrow q$ , what are the converse, inverse and contrapositive?
Answer
Converse: $q \Rightarrow p$ . Inverse: ${\sim}p \Rightarrow {\sim}q$ . Contrapositive: ${\sim}q \Rightarrow {\sim}p$ .
From the lesson: Statements and implications
Rule
Which statement means the same as $p \Rightarrow q$ ?
Answer
Its contrapositive, ${\sim}q \Rightarrow {\sim}p$ . The converse, $q \Rightarrow p$ , does not.
The four implicationsThe gold pair mean the same: p ⇒ q and its contrapositive. The dashed pair mean the same as each other.
From the lesson: Statements and implications
Rule
Draw "All P are Q" as a Venn diagram.
Answer
The circle $P$ inside the circle $Q$ .
All P are QP inside Q (also: only Qs are Ps)
From the lesson: Venn diagrams and valid conclusions
Rule
Draw "No P is Q" as a Venn diagram.
Answer
Two circles that don't meet.
No P is QThe circles don't meet
From the lesson: Venn diagrams and valid conclusions
Rule
Draw "Some P are Q" as a Venn diagram.
Answer
Overlapping circles, with at least one member in the overlap.
Some P are QThe overlap has at least one member
From the lesson: Venn diagrams and valid conclusions
Know it
"Only members may vote." Which circle goes inside which?
Answer
Voters inside members: everyone who votes is a member. The word after "only" is the bigger set.
From the lesson: Venn diagrams and valid conclusions
Rule
All A are B, and all B are C. What follows?
Answer
All A are C: A sits inside B, which sits inside C.
All A are B; all B are CA inside B inside C, so all A are C
From the lesson: Venn diagrams and valid conclusions
Which method?
WAEC 2022 · Paper 2 · Q7 (b)
Consider the following statements:

$P$ : There is no Mathematics student who is not clever. $Q$ : Every Physics student studies Mathematics.

(i) Draw a Venn diagram to represent the statements. (ii) Deduce whether the following statements are valid or not valid. (I) Every clever student is a Physics student. (II) All Physics students are clever. (III) Every Mathematics student studies Physics.
What does "There is no Mathematics student who is not clever" mean?
Answer
Every Mathematics student is clever: the Maths circle inside Clever. "Every Physics student studies Mathematics" then puts Physics inside Maths.
From the lesson: Venn diagrams and valid conclusions Try the question

p	∼p
T	F
F	T

p	q	p ∧ q
T	T	T
T	F	F
F	T	F
F	F	F

p	q	p ∨ q
T	T	T
T	F	T
F	T	T
F	F	F

p	q	p ⇒ q
T	T	T
T	F	F
F	T	T
F	F	T