phil

The table completely exhausts all the possibilities available within the logic of the situation. In otherwords, we have generated sixteen ordered 4-tuples. Applying the expression to the ordered pairs we constructed initially, we refer to them as 2-tuples.
 Propositions of predicate logic can be translated into set-theoretical symbols without any loss in meaning. Since any predicate or property determines a set, a monadic formula such as f(x), where the possession of f determines a set s, can be rewritten as x s. Thus since if in f(x1...x2) determines s we can rewrite the whole formula as (x1...x2) s. Logicians such as Russell and whitehead have fruitfully exploited the ramifications of set theory into other domains to set up an expressive theory of relations.

Elementary Postulates of the Algebra of Sets.

 Following upon what we stated about the translatability of predicate logic into set-theoretic terms, some basic postulates of set theory clearly resemble the rules of inference in propositional logic. We shall list fourteen of these postulates and illustrate a few of them with venn diagram. But before we go into all that, two other operations on sets have to be introduced. The two operations are difference and exclusive union.
 Let s and t be any two subsets of the universal set E. The difference between s and t, written as s-t, is the subset that consists of those elements which are members of s but are not numbers of t. The difference s-t is clearly exhibited in the diagram below.

 The exclusive union of s and t, denoted by sUt is the set of elements of the universal set that belong to s or t but not both. The following venn Diagram illustrates sUt.

 On closer inspection, it can be observed that s-t and sUt can be derived from the three operations on sets namely union, intersection and complementation.
Thus s-t can be expressed as s n t, while 5 u t can be penned down as (s n t) n (s u t) or as (t n s) u (s n t). Returning now to the postulates we undumbrated earlier, a few of interest to the student of Symbolic logic. They include: Postulate 1: s U t
Postu|ate 2:5 n t
Postu|ate 3: s u (t U u) = (s U t) U u
Associative Law
Postu|ate 4: s n (t n u) = (s n t) n u
Postu|ate 5: s U (t n u) = (s U t) n (s U u)
Distributive Law
Postulate 6: s n (t U u) = (s n t) U (s n u)
Postu|ate 7: s n s = s
Postu|ate 8: s U s = s
Postu|ate 9: s U Q = s
Postu|ate 10: s n E = s
Postu|ate 11: s U s = E
Postu|ate 12: s n s = Q
Postulate 13: (s U t) = s n t
De Morgans law
Postu|ate 14:(s n t) = s U t

Postulates 10 down to 12, rely on the definition of the empty set Q universal set E and the complement set. They are exhibited clearly in this Venn Diagram:

 This diagram illustrates clearly that the intersection of the set s and the universal E (that is, S n E) is precisely covered by the area enclosed within the circle. This This diagram illustrates clearly that the intersection of the set s and the universal E (that is, S n E) is precisely covered by the area enclosed within the circle. This establishes postulate 10. Further, the union of s and 5 includes the space within the circle and the space outside the circle enclosed in the boundaries of ABCD, that is within the universal set E. But these spaces are all within E. Therefore, S U S equals E. Finally, S is the complement set of S. Our diagram does not contain any space corresponding to the intersection of
f s and its complementary set. This means that s intersection 5 represents an empty or null set. The null set, you would recall, has no members at all. From the diagram then, it follows that nothing belong to both 5 and s. The use of the set algebra to illustrate sets is demonstrated further in the Venn Diagram below:

 This figure represents the universal set and two subsets within it. Different spaces in the diagram are demarked by the letters a, b, c and u. Those areas or spaces consisting of a, b, c, u make up the universal set E. The set 5 consists of a and b whereas b and c make up set t. All the possible subsets contained in the Venn Diagram can be constructed using the operations on sets earlier introduced:

SetSpace
as n t v s - t
bs nt
csntvt-s
u  (sUt)
a, bs
a,c(snt) U (s nt)n(sUt),v(sUt),v(s-t)U (t-s)
a,ut
b,ct c,u s
a, b, cs U t
a, c, u (s n t)
a, b, c, u E

Example 1Determine w
which of the following statements are true and which are false.

(a) D E {A, B, C, E, F}
(b) A E {1, 2, 3, 4}
(c) F E {2, 4, 6, 8, 10
(d) {1}E]

Solution

(a) False    (b) False    (c) True    (d) False

Example 2

(a) {1,2, 3, 4} U {2, 3, 5} =
(b) {1, 2, 3, 4} n {4, 5, 3} =
(c) {5, 6, 7, 8} U {2, 3, 5} =
(d) {2, 4, 6, 8} n {5, 6, 7} =
(e) {1, 2, 5, 6} n {A, B, C} =

Solution

br>
(a) 1, 2, 3, 4, 5   (b) 3, 4  (c) 2, 3, 5, 6, 7, 8
(d) 6   (e) Q

Example 3

Enumerate the subsets of 1, 2, 3, 4, 5 which have exactly two numbers.

1, 2, 1, 3, 1, 4, 1, 5, 2,3, 2, 4, 2, 5, 3, 4, 3, 5

Example 4

(a) Emumerate the subsets of 1, 2, 3
(b) Which of them are proper subsets of 1, 2, 3

So|ution

(a) 1, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 3, Q
(b) 1, 2, 3, 1, 2, 1, 3, 2, 3, Q

Example 5

For the sets 5 = a, b, c, d and t = b, d, e, determine the membership of s U t, s n t, s U s, s n Q, s U Q,

Solution

s U t = a, b, c, d,e;snt=b,d;sUs=s;sns=s;an=Q;sU Q: s.

Relations

 Although relational propositions could be interpreted in set - theoretical terms, there are well - developed alternatives for articulating and symbolizing such propositions. Relational propositions differ from subject-predicate propositions in that the latter links a subject and predicate together through a copula whereas as the latter does not. Really available illustrations of relational propositions are contained in phrases "to the east of", "on top of", "greater than" etc. In the statement "Obi is older than Ada", or "Aba is to the east of Owerri", Obi or Aba is the referent, and Ada or Owerri is the relatum.