Discrete Mathematics
WEN-CHING LIEN Department of Mathematics National Cheng Kung University
2008
WEN-CHINGLIEN Discrete Mathematics
2.1: Basic Connective and Truth Tables
Statement:
Example-
p :Combinatorics is a required course for sophomores.
q :Margaret Mitchell wrote Gone with the Wind.
1 ”¬p”:not p.
2 (a) ”p∧q”: p and q.(Conjunction) (b) ”p∨q”: p or q.(Disjunction)
(inclusive or)=”and/or”
”p⊻q”: (exclusive or)=”or,but not both.”
(c) ”p→q”:p implies q.(Implication) (If p , then q)
(d) ”p↔q”: p if and only if q.(Biconditional)
2.1: Basic Connective and Truth Tables
Statement:
Example-
p :Combinatorics is a required course for sophomores.
q :Margaret Mitchell wrote Gone with the Wind.
1 ”¬p”:not p.
2 (a) ”p∧q”: p and q.(Conjunction) (b) ”p∨q”: p or q.(Disjunction)
(inclusive or)=”and/or”
”p⊻q”: (exclusive or)=”or,but not both.”
(c) ”p→q”:p implies q.(Implication) (If p , then q)
(d) ”p↔q”: p if and only if q.(Biconditional)
WEN-CHINGLIEN Discrete Mathematics
2.1: Basic Connective and Truth Tables
Statement:
Example-
p :Combinatorics is a required course for sophomores.
q :Margaret Mitchell wrote Gone with the Wind.
1 ”¬p”:not p.
2 (a) ”p∧q”: p and q.(Conjunction) (b) ”p∨q”: p or q.(Disjunction)
(inclusive or)=”and/or”
”p⊻q”: (exclusive or)=”or,but not both.”
(c) ”p→q”:p implies q.(Implication) (If p , then q)
(d) ”p↔q”: p if and only if q.(Biconditional)
2.1: Basic Connective and Truth Tables
Statement:
Example-
p :Combinatorics is a required course for sophomores.
q :Margaret Mitchell wrote Gone with the Wind.
1 ”¬p”:not p.
2 (a) ”p∧q”: p and q.(Conjunction) (b) ”p∨q”: p or q.(Disjunction)
(inclusive or)=”and/or”
”p⊻q”: (exclusive or)=”or,but not both.”
(c) ”p→q”:p implies q.(Implication) (If p , then q)
(d) ”p↔q”: p if and only if q.(Biconditional)
WEN-CHINGLIEN Discrete Mathematics
2.1: Basic Connective and Truth Tables
Statement:
Example-
p :Combinatorics is a required course for sophomores.
q :Margaret Mitchell wrote Gone with the Wind.
1 ”¬p”:not p.
2 (a) ”p∧q”: p and q.(Conjunction) (b) ”p∨q”: p or q.(Disjunction)
(inclusive or)=”and/or”
”p⊻q”: (exclusive or)=”or,but not both.”
(c) ”p→q”:p implies q.(Implication) (If p , then q)
(d) ”p↔q”: p if and only if q.(Biconditional)
2.1: Basic Connective and Truth Tables
Statement:
Example-
p :Combinatorics is a required course for sophomores.
q :Margaret Mitchell wrote Gone with the Wind.
1 ”¬p”:not p.
2 (a) ”p∧q”: p and q.(Conjunction) (b) ”p∨q”: p or q.(Disjunction)
(inclusive or)=”and/or”
”p⊻q”: (exclusive or)=”or,but not both.”
(c) ”p→q”:p implies q.(Implication) (If p , then q)
(d) ”p↔q”: p if and only if q.(Biconditional)
WEN-CHINGLIEN Discrete Mathematics
Truth Table (1) Table 2.1
p ¬q
0 1
1 0
(2)Table 2.2
p q p∧q p∨q p⊻q p→q p ↔q
0 0 0 0 0 1 1
0 1 0 1 1 1 0
1 0 0 1 1 0 0
1 1 1 1 0 1 1
0-false, 1-true Remark:
1. p∧q is true only when p , q are true.
p∨q is false only when both are false.
2.p→q is false except that p is true and q is false.
Truth Table (1) Table 2.1
p ¬q
0 1
1 0
(2)Table 2.2
p q p∧q p∨q p⊻q p→q p ↔q
0 0 0 0 0 1 1
0 1 0 1 1 1 0
1 0 0 1 1 0 0
1 1 1 1 0 1 1
0-false, 1-true Remark:
1. p∧q is true only when p , q are true.
p∨q is false only when both are false.
2.p→q is false except that p is true and q is false.
WEN-CHINGLIEN Discrete Mathematics
Truth Table (1) Table 2.1
p ¬q
0 1
1 0
(2)Table 2.2
p q p∧q p∨q p⊻q p→q p ↔q
0 0 0 0 0 1 1
0 1 0 1 1 1 0
1 0 0 1 1 0 0
1 1 1 1 0 1 1
0-false, 1-true Remark:
1. p∧q is true only when p , q are true.
p∨q is false only when both are false.
2.p→q is false except that p is true and q is false.
Definition (2.1)
A compound statement is called a tautology if it is true for all truth value assignments for its component statements.
If a compound statement is false for all such assignments, then it is called a contradiction.
Exercise:Table 2.3
p q r ¬r ¬r →p q∧(¬r →p)
0 0 0 1 0 0
0 0 1 0 1 0
0 1 0 1 0 0
0 1 1 0 1 1
1 0 0 1 1 0
1 0 1 0 1 0
1 1 0 1 1 0
1 1 1 0 1 1
WEN-CHINGLIEN Discrete Mathematics
Definition (2.1)
A compound statement is called a tautology if it is true for all truth value assignments for its component statements.
If a compound statement is false for all such assignments, then it is called a contradiction.
Exercise:Table 2.3
p q r ¬r ¬r →p q∧(¬r →p)
0 0 0 1 0 0
0 0 1 0 1 0
0 1 0 1 0 0
0 1 1 0 1 1
1 0 0 1 1 0
1 0 1 0 1 0
1 1 0 1 1 0
1 1 1 0 1 1
Definition (2.1)
A compound statement is called a tautology if it is true for all truth value assignments for its component statements.
If a compound statement is false for all such assignments, then it is called a contradiction.
Exercise:Table 2.3
p q r ¬r ¬r →p q∧(¬r →p)
0 0 0 1 0 0
0 0 1 0 1 0
0 1 0 1 0 0
0 1 1 0 1 1
1 0 0 1 1 0
1 0 1 0 1 0
1 1 0 1 1 0
1 1 1 0 1 1
WEN-CHINGLIEN Discrete Mathematics
