15CS3
DISCRETE MATHEMATICAL STRUCTURES 6
Module 1: Set Theory:
Sets and Subsets,
Set Operations and the Laws of Set Theory,
Counting and Venn Diagrams,
A First Word on Probability,
Countable and
Uncountable Sets
Fundamentals of Logic:
Basic Connectives and Truth Tables,
Logic Equivalence –The Laws of Logic,
Logical Implication – Rules of Inference.
DEPT. OF CSE, ACE Page 4
, 15CS3
DISCRETE MATHEMATICAL STRUCTURES 6
Set Theory:
Sets and Subsets:
A set is a of objects, called elements the set. set b
collection of A can be y listing
betwee braces: A = {1, 2, 3, 4, 5}. The symbol belongs
its elements n e is (or to) a set.
3 e A. Its negation is represented e.g. finite,
For instance by /e, 7 /e A. If the set Is its
number of elements is represented |A|, e.g. if A = {1, 2, 3, | A| =
4, 5} then 5
1. N = {0, 1, 2, 3, ·· · } = the set of natural numbers.
2. Z = {··· , -3, -2, -1, 0, 1, 2, 3, ··· } = the set of integers.
3. Q = the set of rational numbers.
4. R = the set of real numbers.
5. C = the set of complex numbers.
If S is one of then we also use the following
those sets notations :
1. S = element in S, for
+
set of positive s instance
Z
= {1, 2, 3, the set of positive
+
··· } = integers.
= of negative
-
2. S set elements in S, for instance
= {-1, -2, -3, set of negative
-
Z ··· } = the integers.
∗
3. S =
of in excluding zero, for
∗
set elements S instance R = the set of non zero real
num
bers.
Set-builder An way to define a called set-
notation: alternative set, builder notation, is
by propert (predicat verifie by its
stating a y e) P (x) d exactly elements,for instance
A = {x e | 1 ≤ x ≤ 5} “set of x such that 1 ≤ ≤ 5”—
Z = integers x i.e.: A = {1, 2, 3,
4, general: A = {x e U | p(x)}, U univers of
5}. In where is the e discourse in which
the mus be interpreted, or A = {x | P (x)} if the universe of
predicate P (x) t discourse
for P (x) is understood In set the term universalis often used in
, implicitly . theory set
place of “universe of discourse” for a given
predicate.
Princip of Extension: Two sets are equal only if
le if and they have the same
A = � ∀x (x e A ↔ x e B)
elements, i.e.: B .
Subset: We say A is a of set or A is in B,
that subset B, contained and we represent
if all of A if A = {a, b, c}
it “A ⊆ B”, elements are in B, e.g., and
B = {a, b, c, d, e} then A ⊆
B.
Proper subset: prope subse of represente
A is a r t B, d “A ⊂ B”, if A ⊆ B
i.e., there is some element which is
A = B, in B not in A.
DEPT. OF CSE, Page
ACE 5
, 15CS3
DISCRETE MATHEMATICAL STRUCTURES 6
Empty Set: A set with no elements is called empty set
(or null set, or void set ), and is represented by
∅ or {}.
Note that preven a set from possibly element of se (whic
nothing ts being an another t h
is not the same as being a
subset!). For i n stance
is an elem ent of
if A {1, a, {3, t}, { 1, 2, 3}} { 3, t}, t obvious ly
= and B= hen B A,
i.e., e
B A.
Pow Set: collectio of subse A is the power set of
er The n all ts of a set called A,
is P(A). For instance, if A = {1, 2, 3},
and represented then
P(A) = {∅, {2}, {3}, {1, 2}, {1, 3}, {2, 3},
{1}, A} .
MultCSE ordinar set identical if they have same
ts: Two ys are the elements, so for
instance, {a, a, b} {a, b} the set because exactl
and are same they have y the same
elements namel Howeve in application it might
, y a and b. r, some s be useful to
element a w us multCSEt ar
allow repeated s in set. In that case e e s, which e
mathematical entities similar to possibl repeat elements So
sets, but with y ed . , as
multCSEts, {a, a, b} and {a, b} would be consi dered since in the first one
different, the
element a occurs twice in the second one it occurs only
and once.
S et Oper atio ns:
1. Intersection : The common elements of two sets:
A ∩ B = {x | (x e A) ∧ (x e B)} .
If A ∩ B = ∅, the sets are said to be disjoint.
2. Union : The set of elements that belong to either of two sets:
A ∪ B = {x | (x e A) ∨ (x e B)} .
DISCRETE MATHEMATICAL STRUCTURES 6
Module 1: Set Theory:
Sets and Subsets,
Set Operations and the Laws of Set Theory,
Counting and Venn Diagrams,
A First Word on Probability,
Countable and
Uncountable Sets
Fundamentals of Logic:
Basic Connectives and Truth Tables,
Logic Equivalence –The Laws of Logic,
Logical Implication – Rules of Inference.
DEPT. OF CSE, ACE Page 4
, 15CS3
DISCRETE MATHEMATICAL STRUCTURES 6
Set Theory:
Sets and Subsets:
A set is a of objects, called elements the set. set b
collection of A can be y listing
betwee braces: A = {1, 2, 3, 4, 5}. The symbol belongs
its elements n e is (or to) a set.
3 e A. Its negation is represented e.g. finite,
For instance by /e, 7 /e A. If the set Is its
number of elements is represented |A|, e.g. if A = {1, 2, 3, | A| =
4, 5} then 5
1. N = {0, 1, 2, 3, ·· · } = the set of natural numbers.
2. Z = {··· , -3, -2, -1, 0, 1, 2, 3, ··· } = the set of integers.
3. Q = the set of rational numbers.
4. R = the set of real numbers.
5. C = the set of complex numbers.
If S is one of then we also use the following
those sets notations :
1. S = element in S, for
+
set of positive s instance
Z
= {1, 2, 3, the set of positive
+
··· } = integers.
= of negative
-
2. S set elements in S, for instance
= {-1, -2, -3, set of negative
-
Z ··· } = the integers.
∗
3. S =
of in excluding zero, for
∗
set elements S instance R = the set of non zero real
num
bers.
Set-builder An way to define a called set-
notation: alternative set, builder notation, is
by propert (predicat verifie by its
stating a y e) P (x) d exactly elements,for instance
A = {x e | 1 ≤ x ≤ 5} “set of x such that 1 ≤ ≤ 5”—
Z = integers x i.e.: A = {1, 2, 3,
4, general: A = {x e U | p(x)}, U univers of
5}. In where is the e discourse in which
the mus be interpreted, or A = {x | P (x)} if the universe of
predicate P (x) t discourse
for P (x) is understood In set the term universalis often used in
, implicitly . theory set
place of “universe of discourse” for a given
predicate.
Princip of Extension: Two sets are equal only if
le if and they have the same
A = � ∀x (x e A ↔ x e B)
elements, i.e.: B .
Subset: We say A is a of set or A is in B,
that subset B, contained and we represent
if all of A if A = {a, b, c}
it “A ⊆ B”, elements are in B, e.g., and
B = {a, b, c, d, e} then A ⊆
B.
Proper subset: prope subse of represente
A is a r t B, d “A ⊂ B”, if A ⊆ B
i.e., there is some element which is
A = B, in B not in A.
DEPT. OF CSE, Page
ACE 5
, 15CS3
DISCRETE MATHEMATICAL STRUCTURES 6
Empty Set: A set with no elements is called empty set
(or null set, or void set ), and is represented by
∅ or {}.
Note that preven a set from possibly element of se (whic
nothing ts being an another t h
is not the same as being a
subset!). For i n stance
is an elem ent of
if A {1, a, {3, t}, { 1, 2, 3}} { 3, t}, t obvious ly
= and B= hen B A,
i.e., e
B A.
Pow Set: collectio of subse A is the power set of
er The n all ts of a set called A,
is P(A). For instance, if A = {1, 2, 3},
and represented then
P(A) = {∅, {2}, {3}, {1, 2}, {1, 3}, {2, 3},
{1}, A} .
MultCSE ordinar set identical if they have same
ts: Two ys are the elements, so for
instance, {a, a, b} {a, b} the set because exactl
and are same they have y the same
elements namel Howeve in application it might
, y a and b. r, some s be useful to
element a w us multCSEt ar
allow repeated s in set. In that case e e s, which e
mathematical entities similar to possibl repeat elements So
sets, but with y ed . , as
multCSEts, {a, a, b} and {a, b} would be consi dered since in the first one
different, the
element a occurs twice in the second one it occurs only
and once.
S et Oper atio ns:
1. Intersection : The common elements of two sets:
A ∩ B = {x | (x e A) ∧ (x e B)} .
If A ∩ B = ∅, the sets are said to be disjoint.
2. Union : The set of elements that belong to either of two sets:
A ∪ B = {x | (x e A) ∨ (x e B)} .
