Discrete Mathematics

1.1 Propositional logic

1.Proposition

1.Def: A declarative sentence that is either true or false ,but not both.

ex1: x+1=2
Sol: This is not a proposition, cuz when x=1,it’s true while other x makes it false.

2.propositional variables : letters like (p,q,r,s) can denote a propositon

2.logic operators:

1.negation operator: $\neg$
2.conjunction: $\wedge$
3.disjunction $\vee$
can be divided into [ exclusive OR and inclusive OR]
exclusive or , either q or p can be true ,but not both
inclusive or , either q or p be true ,and namely they can both be true
4.conditional statement:$\to$
ways to express implication
if p then q ; p implies q ; if p ,q ; p only if q ; p is sufficient for q;
q is necessary for p ; q if p ; q whenever p ; q unless $\neg$ p

p only if q may be the most confusing one:
note that p only if q says that p cannot be true when q is not true.That is when p is true ,q is false the statement is wrong. When p is false whatevear q is true ,cuz the statement only cares about the truth value of p.
5.bioconditionals:$\leftrightarrow$
common ways to express it
p is necessary and sufficient for q ; if p then q and conversely;
p if and only if q plus the abbreviation ‘iff’ which can denote only and only if .

3. Precedence of Logic operators

Operator	Precedence
$\neg$	1
$\wedge$	2
$\vee$	3
$\to$	4
$\leftrightarrow$	5

4. Converse ,Inverse and Contrapostive

converse of p$\to$q : q$\to$p
inverse of p$\to$q:$\neg$p$\to \neg$q
contrastive of p$\to$q: $\neg$q $\to$ $\neg$q
The contrastive of a statement is equivalent to itself.

5.Bit and Bitwise operation

OR AND XOR

1.2 Application of Propositional Logic

1.Boolean Search

2.Logic puzzles

3.Digital circuits

Both Title 2 & 3 can be solved by using truth table

1.3 Propositional equivalences

1.tautology and contradiction

Def : a compound proposition that is always true , no matter what the truth value of the propositional variables involved in , is called a tautology .
The other proposition which is always false is called contradiction.

2.Logical equivalence

(1)Def : if p$\leftrightarrow$ q is a tautology , then notation p $\equiv$q denotes that p and q are logically equivalent.

(2)ways of proving logical equivalence :
i.truth table : if and only if the truth table agrees can prove
Be care that n variables need a 2^n truth table

ii.logical reasoning : by using the proved logical equivalent to prove the unknown one.

(3) important logical equivalance
i. Identity law
p$\wedge$T$\equiv$p
p$\vee$F$\equiv$p
ii.Domination law
p$\vee$T$\equiv$T
p$\wedge$F$\equiv$F
iii.Indepotent law
p$\wedge$p$\equiv$p
p$\vee$p$\equiv$p
iv.Double negation law
$\neg \neg$p$\equiv$p
v.Commutative laws
p$\vee$q$\equiv$q$\vee$p
p$\wedge$q$\equiv$q$\wedge$p
vi.Associative laws
(p$\wedge$q)$\wedge$r$\equiv$p$\wedge$(q$\wedge$r)
(p$\vee$q)$\vee$r$\equiv$p$\vee$(q$\vee$r)
vii.Distributive laws
p$\wedge$(q$\vee$r)$\equiv$(p$\wedge$q)$\vee$(p$\wedge$r)
p$\vee$(q$\wedge$r)$\equiv$(p$\vee$q)$\wedge$(p$\vee$r)
viii.De morgan’s laws
$\neg$(p$\vee$q)$\equiv$$\neg$p$\wedge$$\neg$q
$\neg$(p$\wedge$q)$\equiv$$\neg$p$\vee$$\neg$q
ix.Absorption laws
p$\vee$(p$\wedge$q)$\equiv$p
p$\wedge$(p$\vee$q)$\equiv$p
x.Negation laws
p$\wedge$$\neg$p$\equiv$F
p$\vee$$\neg$p$\equiv$T
Conditional logic equivalence
xi.
p$\to$q$\equiv$$\neg$p$\vee$q
xii.
p → q ≡ ¬p ∨ q
xiii.
p → q ≡ ¬q → ¬p
xiv.
p ∨ q ≡ ¬p → q
xv.
p ∧ q ≡ ¬(p → ¬q)
xvi.
¬(p → q) ≡ p ∧ ¬q
xvii.
(p → q) ∧ (p → r) ≡ p → (q ∧ r)
(p → r) ∧ (q → r) ≡ (p ∨ q) → r
(p → q) ∨ (p → r) ≡ p → (q ∨ r)
(p → r) ∨ (q → r) ≡ (p ∧ q) → r
Bioconditional equivalence
xviii.
p ↔ q ≡ (p → q) ∧ (q → p)
xix.
p ↔ q ≡ ¬p ↔ ¬q
xx.
p ↔ q ≡ (p ∧ q) ∨ (¬p ∧ ¬q)
xxi.
¬(p ↔ q) ≡ p ↔ ¬q

3.Extending use of associative laws

We can use the notation Vⁿ_j=1p_j for p₁V p₂ V p₃ V…V p_n
and $\wedge$ⁿ_j=1 as well.

4.Proposition Satisfiablity