Discrete Mathematics
WEN-CHING LIEN
Department of Mathematics National Cheng Kung University
2008
2.4: The use of Quantifiers
Definition (2.5)
A declarative sentence is an open statement if 1) it contains one or more variables, and 1’) quantifier:
1 ∃x (the existential quantifier)
2 ∀x (the universal quantifier)
2) it is not a statement, but
3) it becomes a statement when the variables in it are replaced by certain allowable choices.
2.4: The use of Quantifiers
Definition (2.5)
A declarative sentence is an open statement if 1) it contains one or more variables, and 1’) quantifier:
1 ∃x (the existential quantifier)
2 ∀x (the universal quantifier)
2) it is not a statement, but
3) it becomes a statement when the variables in it are replaced by certain allowable choices.
2.4: The use of Quantifiers
Definition (2.5)
A declarative sentence is an open statement if 1) it contains one or more variables, and 1’) quantifier:
1 ∃x (the existential quantifier)
2 ∀x (the universal quantifier)
2) it is not a statement, but
3) it becomes a statement when the variables in it are replaced by certain allowable choices.
2.4: The use of Quantifiers
Definition (2.5)
A declarative sentence is an open statement if 1) it contains one or more variables, and 1’) quantifier:
1 ∃x (the existential quantifier)
2 ∀x (the universal quantifier)
2) it is not a statement, but
3) it becomes a statement when the variables in it are replaced by certain allowable choices.
2.4: The use of Quantifiers
Definition (2.5)
A declarative sentence is an open statement if 1) it contains one or more variables, and 1’) quantifier:
1 ∃x (the existential quantifier)
2 ∀x (the universal quantifier)
2) it is not a statement, but
3) it becomes a statement when the variables in it are replaced by certain allowable choices.
Definition (2.6)
Let p(x),q(x)be open statements defined for a given universe.
The open statements p(x)and q(x)are called (logically) equivalent,and
we write∀x[p(x) ⇔q(x)]when the biconditional p(a) ↔q(a)is true for each replacement a from the universe (that is,
p(a) ⇔q(a)for each a in the universe).
If the implication p(a) →q(a)is true for each a in the universe(that is, p(a) ⇒q(a)for each a in the universe), then we write∀x[p(x) ⇒q(x)]and say that p(x)logically implies q(x).
Definition (2.7)
For open statements p(x),q(x)–defined for a prescribed universe – and the universally quantified statement
∀x[p(x) →q(x)], we define:
1 The contrapositive of∀x[p(x) →q(x)]to be
∀x[¬q(x) → ¬p(x)]
2 The converse of∀x[p(x) →q(x)]to be∀x[q(x) →p(x)]
3 The inverse of∀x[p(x) →q(x)]to be∀x[¬p(x) → ¬q(x)]
Definition (2.7)
For open statements p(x),q(x)–defined for a prescribed universe – and the universally quantified statement
∀x[p(x) →q(x)], we define:
1 The contrapositive of∀x[p(x) →q(x)]to be
∀x[¬q(x) → ¬p(x)]
2 The converse of∀x[p(x) →q(x)]to be∀x[q(x) →p(x)]
3 The inverse of∀x[p(x) →q(x)]to be∀x[¬p(x) → ¬q(x)]
Definition (2.7)
For open statements p(x),q(x)–defined for a prescribed universe – and the universally quantified statement
∀x[p(x) →q(x)], we define:
1 The contrapositive of∀x[p(x) →q(x)]to be
∀x[¬q(x) → ¬p(x)]
2 The converse of∀x[p(x) →q(x)]to be∀x[q(x) →p(x)]
3 The inverse of∀x[p(x) →q(x)]to be∀x[¬p(x) → ¬q(x)]
Definition (2.7)
For open statements p(x),q(x)–defined for a prescribed universe – and the universally quantified statement
∀x[p(x) →q(x)], we define:
1 The contrapositive of∀x[p(x) →q(x)]to be
∀x[¬q(x) → ¬p(x)]
2 The converse of∀x[p(x) →q(x)]to be∀x[q(x) →p(x)]
3 The inverse of∀x[p(x) →q(x)]to be∀x[¬p(x) → ¬q(x)]
Example (2.41)
p(x) : |x| >3 q(x) :x >3 r(x) :x <−3
Statement: ∀x[p(x) → (r(x) ∨q(x))]
Contrapositive: ∀x[¬(r(x) ∨q(x)) → ¬p(x)]
Converse: ∀x[(r(x) ∨q(x)) →p(x)]
Inverse: ∀x[¬p(x) → ¬(r(x) ∨q(x))]
The statement
∀x[p(x) → (r(x) ∨q(x))]
Example (2.41)
p(x) : |x| >3 q(x) :x >3 r(x) :x <−3
Statement: ∀x[p(x) → (r(x) ∨q(x))]
Contrapositive: ∀x[¬(r(x) ∨q(x)) → ¬p(x)]
Converse: ∀x[(r(x) ∨q(x)) →p(x)]
Inverse: ∀x[¬p(x) → ¬(r(x) ∨q(x))]
The statement
∀x[p(x) → (r(x) ∨q(x))]
Example (2.41)
p(x) : |x| >3 q(x) :x >3 r(x) :x <−3
Statement: ∀x[p(x) → (r(x) ∨q(x))]
Contrapositive: ∀x[¬(r(x) ∨q(x)) → ¬p(x)]
Converse: ∀x[(r(x) ∨q(x)) →p(x)]
Inverse: ∀x[¬p(x) → ¬(r(x) ∨q(x))]
The statement
∀x[p(x) → (r(x) ∨q(x))]
Example (2.41)
p(x) : |x| >3 q(x) :x >3 r(x) :x <−3
Statement: ∀x[p(x) → (r(x) ∨q(x))]
Contrapositive: ∀x[¬(r(x) ∨q(x)) → ¬p(x)]
Converse: ∀x[(r(x) ∨q(x)) →p(x)]
Inverse: ∀x[¬p(x) → ¬(r(x) ∨q(x))]
The statement
∀x[p(x) → (r(x) ∨q(x))]
Example (2.41)
p(x) : |x| >3 q(x) :x >3 r(x) :x <−3
Statement: ∀x[p(x) → (r(x) ∨q(x))]
Contrapositive: ∀x[¬(r(x) ∨q(x)) → ¬p(x)]
Converse: ∀x[(r(x) ∨q(x)) →p(x)]
Inverse: ∀x[¬p(x) → ¬(r(x) ∨q(x))]
The statement
∀x[p(x) → (r(x) ∨q(x))]
Example (2.41)
p(x) : |x| >3 q(x) :x >3 r(x) :x <−3
Statement: ∀x[p(x) → (r(x) ∨q(x))]
Contrapositive: ∀x[¬(r(x) ∨q(x)) → ¬p(x)]
Converse: ∀x[(r(x) ∨q(x)) →p(x)]
Inverse: ∀x[¬p(x) → ¬(r(x) ∨q(x))]
The statement
∀x[p(x) → (r(x) ∨q(x))]
Example (2.41)
p(x) : |x| >3 q(x) :x >3 r(x) :x <−3
Statement: ∀x[p(x) → (r(x) ∨q(x))]
Contrapositive: ∀x[¬(r(x) ∨q(x)) → ¬p(x)]
Converse: ∀x[(r(x) ∨q(x)) →p(x)]
Inverse: ∀x[¬p(x) → ¬(r(x) ∨q(x))]
The statement
∀x[p(x) → (r(x) ∨q(x))]
