Lecture : Set Theory and Logic

Transcription

1 Lecture : Dr. Department of Mathematics Lovely Professional University Punjab, India October 18, 2014

2 Outline Contrapositive and Converse 1 Contrapositive and Converse

3 Contrapositive and Converse I Our discission of the if... then construction leads to us to consider another point of elementary logic that some times causes difficulty. It concern the relation between a statement, its contrapositive, and its converse. Given a statement of the form If P, then Q, its contrapositive is defined to the statement If Q is not true. For example, the contrapositive of the statement If x > 0, then x 3 0 is the statement If x 3 = 0, then it is not true that x > 0 Here both the statement and its contrapositive are true.

4 Contrapositive and Converse II The statement If x 2 < 0, then x = 23 has as its contrapositive the statement If x = 23, then it is not true that x 2 < 0 Again, both are true statements about real numbers.

5 Contrapositive and Converse III From these two example one question aries that Question : There is some ration between a statement and its contrapositive? Answer : These two ways of saying precisely same thing. Each is true if an only if other is true. They are logically equivalent. Now how you will show that both are same.

6 Contrapositive and Converse IV Let us introduce some notation If P, then Q will be write as P Q The contrapositive can then be expressed in the form (not Q) (not P )

7 Contrapositive and Converse V There is another statement that can be formed from the statement P Q. It is the statement Q P Which is called the converse of P Q. Remark Careful about the difference between a statement s converse and contrapositive. Whereas a statement and its contrapositive are logically equivalent, the truth of a statment say noting at all about the truth or falsity of its converse.

8 Contrapositive and Converse VI The true statement If x > 0, then x 3 0 has as its converse the statement which is false. Similarly, the true statement has as its converse the statement If x 3 0, then x > 0 If x 2 < 0, then x = 23 If x = 23, then x 2 < 0 which is false.

9 Contrapositive and Converse VII Remark Both the statement P Q and its converse Q P are true, we express this fact by the notation P Q

10 I Definition If one wishes to form the contrapositive of the statement P Q, one has to know how to form the statement not P, which is called the negation of P. Consider an example, suppose that X is a set, A is a subset of X, and P is a statement about the general element of X. Consider the following statement For every x A, statement P holds (1) Then negation of the statement will be For at least one x A, statement P does not hold (2)

11 II Therefore, to form the negation of the statement, one replace the quantifier for every by quantifier for at least, and one replaces statement P by its negation. The process works in reverse just as well the negation of the statement is the statement For at least one x A, statement Q holds (3) For every x A, statement Q does not hold (4)

12 I When we have a set whose elements are sets often refer to it as a collection of sets and denote it by a script letter such that as A of B. Take care on this, if A is the set {a, b, c}, then the statments are all true. a A, {a} A and {a} P(A)

13 I Given a collection A of sets, the union of the elements of A is defined by the equation A A A = {x: x A for at least one A A} (5) The intersection of the elements of A is defined by the equation A A A = {x: x A for every A A} (6)

14 I If A and B are two sets, the their cartesian product is defined as A B = {(a, b): a A and b B}