Logic General

Introduction of logic general:

The field includes both the mathematical study of logic general and the applications of formal logic to other areas of mathematics. The unifying themes in mathematical logic general include the study of the expressive power of formal systems and the deductive power of formal proof systems.
Mathematical logic general is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share and basic results on logic, particularly first-order logic, and definably.
(Source from Wikipedia)

Mathematical Logic General Divided into Four Important Parts:

1)Set theory

2)Model theory

3)Recursion theory, and

4)Proof theory and constructive mathematics

Logic general for set theory:

If an element x belongs to a set A, we write x ? A. A set containing all the objects likely to be considering in a par8+ticular discussion is called a universal set and is usually denoted by U.

Logic general for proof theory:

A logical argument to be establishes the validity of a proposition or mathematical formula. The proof theory comprised if a set of axioms and premises sequentially to arrive at a formula to be proved in the form of a conclusion.

List of some Logical general operators:

( )           brackets

!  + -        logical not, unary plus, unary minus

* / %         multiply, divide, modulus

+ -          add, subtract

< <=         less than, less than or equal,

> >=         greater than, greater than or equal

==  !=       equal, not equal

&&           logical and

||             logical or

=             assignment

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Examples of Set Theory - Logic General:

Example 1:

Consider the sets:
f, A = { 1, 3 }, B = {1, 5, 9}, C = {1, 3, 5, 7, 9}.
Insert the symbol ? or ? between each of the following pair of sets:
(i) f . . . B (ii) A . . . B (iii) A . . . C (iv) B . . . C

Solution:

(i) f ? B as f is a subset of every set.
(ii) A ? B as 3 ? A and 3 ? B
(iii) A ? C as 1, 3 ? A also belongs to C
(iv) B ? C as each element of B is also an element of C.

Example 2:

Let A, B and C be three sets. If A ? B and B ? C, is it true that A ? C?. If not, give an example.

Solution:

No, Let A = {1}, B = {{1}, 2} and C = {{1}, 2, 3}. Here A ? B as A = {1}

and B ? C. But A ? C as 1 ? A and 1 ? C.
Note that an element of a set can never be a subset of itself.