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Material Type: Notes; Class: Mathematics: A Human Endeavor; Subject: Mathematics; University: Eastern Illinois University; Term: Spring 2005;
Typology: Study notes
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v p → q is equivalent to (∼ p ∨ q)
v The negation of p → q is (p ∧ ∼ q)
v We can use Truth Tables to show two conditional expressions are equivalent (their truth values will be the same)
v A tautology is a statement which is always TRUE.
v Some circuits can be simplied.
~p p
~p
p
r
p q
p
Direct Statement p → q If p, then q Converse q → p If q, then p Inverse ∼ p →∼ q If not p, then not q Contrapositive ∼ q →∼ p If not q, then not p
Let p = ìthey stayî and q = ìwe leaveî Direct Statement (p → q):
Converse :
Inverse :
Contrapositive :
Direct Converse Inverse Contrapositive p → q q → p ∼ p → ∼ q ∼ q → ∼ p p q ∼ p ∨ q
T T T T
T F F T
F T T F
F F T T
→ 4 is equivalent to ∼ ∨ 4
∼ ∨ 4 ≡ → 4
∨ 4 ≡ ∼ → 4
For p ∨ q, write each of the following:
Direct Statement :
Converse :
Inverse :
Contrapositive :
You'll be sorry if I go.
Today is Thursday only if yesterday was Wednesday.
All nurses wear white shoes.
A stitch in time saves nine.
Rolling stones gather no moss.
Birds of a feather ock together.
Two statements about the same object are: consistent ó if they are both true. contrary ó if they cannot both be true.
∼ p negation of p truth value is opposite of p
p ∧ q conjunction true only when both p and q are true
p ∨ q disjunction false only when both p and q are false
p → q conditional false only when p is true and q is false
p ↔ q biconditional true only when p and q have the same truth value.