set theory 3.pptx
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The document discusses different types of sets: - Finite sets have a one-to-one correspondence with a natural number n representing their cardinality. - Infinite sets are not finite and can be countably or uncountably infinite. Countably infinite sets have a one-to-one correspondence with the natural numbers. - The Cartesian product of sets A and B is the set of all ordered pairs (a,b) where a is in A and b is in B. - The principle of inclusion and exclusion states that the size of the union of sets A1 and A2 is the sum of their individual sizes minus their intersection size.
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A set is a well defined collection of objects, called the “elements” or “members” of the set.
A specific set can be defined in two ways-
If there are only a few elements, they can be listed individually, by writing them between curly braces ‘{ }’ and placing commas in between. E.g.- {1, 2, 3, 4, 5}
The second way of writing set is to use a property that defines elements of the set.
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If x is an element o set A, it can be written as ‘x A’
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The four basic operations are:
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- 2. Finite sets • A set is to be said finite if there is one-to-one correspondence between the elements in the set and the elements in some set n. where n ∈ N n is said cardinality of a set.
- 3. Infinite sets • A set is an infinite set if it is not finite. • Two types of infinite sets – Countably infinite set – Uncountably infinite set { ex. R, } • A set is to be said countably infinite if there is one-to-one correspondence between the elements in the set and the elements in set N. • ex . set of odd positive integers is a countable infinite set.
- 4. Cartesian Product • Let A and B be sets. The Cartesian product of A and B, denoted by A × B, is the set of all ordered pairs (a, b), where a ∈ A and b ∈ B. Hence, • A × B = {(a, b) | a ∈ A ∧ b ∈ B}. • What is the Cartesian product of A = {1, 2} and B = {a, b, c}? Solution: The Cartesian product A × B is A × B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}.
- 5. Principle of inclusion and exclusion • |A1 ∪ A2| = |A1| + |A2| − |A1 ∩ A2|
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