logic mathematics-sub field of mathematics
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LOGIC MATHEMATICS
Statement or proposition: ❖ A Declarative statement which is either
True or False is called a statement in logic ❖ A Statement is also called as proposition LOGICAL CONNECTIVES: ❖ The five terms NOT, OR, AND, if....then, if
and only if are called logical connectives. ❖ A statement in which no connective
appears is called an atomic statements. ❖ A statement in which one or more
connectives appears is called compound statements.
LOGIC MATHEMATICS
DISJUNCTION(OR) p q p ∨ q
T T T
T F T
F T T
F F F
NOTE: If one is true then true else false
Negation (NOT)
p ~p
T F
F T
LOGIC MATHEMATICS
Conjunction (AND)
p q p∧q
T T T
T F F
F T F
F F F NOTE: If both are true then true else false
LOGIC MATHEMATICS
Conditional/Implication(p⟶q)
p q p⟶q
T T T
T F F
F T T
F F T NOTE: If first is true second is false then false else true
LOGIC MATHEMATICS
Biconditional (p↔q)
p q p↔q
T T T
T F F
F T F
F F T Note : Both are same then true else false If p then q p⟶q=(~q⟶~p) =(~p∨q ) p if and only if q p↔q=(p⟶q)∧(q⟶p)
LOGIC MATHEMATICS
❖ Negation(NOT) is an unary connectives
while others are binary connectives because they are combined two statement. ❖ A statement which is alway true is
called Tautology ❖ A statement which is always false is
called Contradiction. ❖ A statement which is not true or false is
Contingency Two statement p and q are said to be logically equally if and only if their truth value are identical. If p⟶q is a conditional statement Then q⟶p is called converse ~p⟶~q is called inverse ~q⟶~p is called contrapositive
LOGIC MATHEMATICS
Demorgan’s Laws: 1) ~(p ∨q)=~p ∧~q 2) ~(p ∧q)=~p ∨~q I Ⅱ
p q ~p ~q (p∨q) ~(p∨q) ~p∧~q
T T F F T F F
T F F T T F F
F T T F T F F
F F T T F T T
From I & Ⅱ ~(p ∨q)= ~p∧~q
LOGIC MATHEMATICS
I Ⅱ
p q ~p ~q p∧q ~(p∧q) ~p∨~q
T T F F T F F
T F F T F T T
F T T F F T T
F F T T F T T
From I & Ⅱ ~(p ∧q)=~p∨~q Hence proved Demorgan’s laws. (from both truth tables)
LOGIC MATHEMATICS
Prove that:
1) (p⟶q)=~q⟶~p=~p∨q I Ⅱ Ⅲ
p q ~p ~q p⟶q ~q⟶~p ~p ∨q
T T F F T T T
T F F T F F F
F T T F T T T
F F T T T T T From I,Ⅱ and Ⅲ (p⟶q)=~q⟶~p=~p∨q
LOGIC MATHEMATICS
2) (p ↔q)=(p ⟶ q)∧(q⟶ p) I Ⅱ
p q (p↔q) (p⟶q) (q⟶p)
(p⟶q)∧(q⟶p)
T T T T T T
T F F F T F
F T F T F F
F F T T T T From I & Ⅱ (p↔q)=(p⟶ q) ∧(q ⟶ p)
LOGIC MATHEMATICS
Solve this problem and write whether it is tautology,contradiction or contingency.
1) (p⟶q)↔(~q⟶~p) A B A↔B
p q ~p ~q p⟶q ~q⟶~p (p⟶q)↔(~q⟶~p)
T T F F T T T T F F T F F T F T T F T T T F F T T T T T
This is tautology
LOGIC MATHEMATICS
2) (p⟶q) ∧(q⟶r)⟶(p⟶r) A B C p q r (p⟶q) (q⟶r) (p⟶r)
T T T T T T T T F T F F T F T F T T T F F F T F F T T T T T F T F T F T F F T T T T F F F T T T
LOGIC MATHEMATICS
A∧B (A∧B)⟶C
(p⟶q)∧(q⟶r) (p⟶q)∧(q⟶r)⟶(p⟶r)
T T F T F T F T T T F T T T T T
This is tautology
LOGIC MATHEMATICS
If we define p↓q to be a true statement of neither p nor q is true Note: Both are false then true else false p q p↓q
T T F T F F F T F F F T
LOGIC MATHEMATICS
Prepare true table for the following 1) (p↓q)∨(p↓r) p q r p↓q p↓r
(p↓q)∨(p↓r) T T T F F F T T F F F F T F T F F F T F F F F F F T T F F F F T F F T T F F T T F T F F F T T T
LOGIC MATHEMATICS
2) (p↓q)↓r p q r (p↓q) (p↓q)↓r
T T T F F T T F F T
T F T F F T F F F T F T T F F F T F F T F F T T F F F F T F
LOGIC MATHEMATICS
We define pΔ q to be a true statement if either p nor q is true but not both. Make true table for the following Note: Both same then false else true p q pΔ q T T F T F T F T T F F F
LOGIC MATHEMATICS
1) (pΔq)Δp p q pΔq
(pΔq)Δp T T F T T F T F F T T T F F F F 2) pΔ~p p ~p pΔ~p T F T F T T
LOGIC MATHEMATICS
3) (pΔq)Δp p q pΔq (pΔq)Δp T T F T T F T F F T T T F F F F
LOGIC MATHEMATICS
4) (pΔq)Δ(qΔr) p q r pΔq T T T F T T F F T F T T T F F T F T T T F T F T F F T F F F F F
LOGIC MATHEMATICS
qΔr (pΔq)Δ(qΔr) F F T T T F F T F T T F T T F F
LOGIC MATHEMATICS
Find the truth values of each statement if p and q are true and r and s,t are false.
1)~(p⟶q ) ~(T⟶T) ~(T) F 2) (~q⟶(r⟶(r⟶(p∨s))) (~T)⟶(F⟶(F⟶(T∨F))) F⟶(F⟶(F⟶T)) F⟶(F⟶T) F⟶T T
LOGIC MATHEMATICS
3) (~p)⟶r (~T)⟶F (F)⟶F T 4) (p⟶s)∧ (s⟶t) (T⟶F)∧(F⟶F) (F)∧(T) F 5) t⟶~q F⟶~(T) F⟶F T
LOGIC MATHEMATICS
6) p⟶(r⟶q) T⟶(F⟶T) T⟶T T 7) (q⟶(r⟶s))∧((p⟶s)⟶(~t)) (T⟶(F⟶F)∧(T⟶F)⟶(~F)) (T⟶T)∧(F⟶(T)) T∧T T 8) (r∧(s∧t))⟶(p∨q) (F∧(F∧F))⟶(T∨T) (F∧F)⟶T F⟶T T