Basics of Sets
... So far in the worksheet, you have seen 8 sets: A, B, C, D, E, F, G and H. Write down the cardinality of each of these sets, using proper mathematical notation to express your answer. Do you think the infinite sets (if there are any) satisfy Cantor’s definition of countably infinite? Write down any r ...
... So far in the worksheet, you have seen 8 sets: A, B, C, D, E, F, G and H. Write down the cardinality of each of these sets, using proper mathematical notation to express your answer. Do you think the infinite sets (if there are any) satisfy Cantor’s definition of countably infinite? Write down any r ...
STD VII - Kerala Samajam Model School
... Identify finite, infinite, empty and the singleton sets among the following: a. Set of odd numbers divisible by 2 b. Set of even prime numbers c. { x: x is right angle in an acute angled triangle} d. The set of positive integers greater than 100 e. { x:x is an obtuse angle in a scalene triangle} f. ...
... Identify finite, infinite, empty and the singleton sets among the following: a. Set of odd numbers divisible by 2 b. Set of even prime numbers c. { x: x is right angle in an acute angled triangle} d. The set of positive integers greater than 100 e. { x:x is an obtuse angle in a scalene triangle} f. ...
WEEK 1: CARDINAL NUMBERS 1. Finite sets 1.1. For a finite set A
... 1.2. If A, B are two finite sets then #(A × B) = #A#̇B. 1.3. P (A), the power set of A, has 2#A elements. To see this, assume A = {1, 2, .., n}. Attach to each subset S ⊂ A the binary string y = y1 y2 ...yn where yi = 1 if i ∈ S and 0 otherwise. Such a correspondence is bijective, and the number of ...
... 1.2. If A, B are two finite sets then #(A × B) = #A#̇B. 1.3. P (A), the power set of A, has 2#A elements. To see this, assume A = {1, 2, .., n}. Attach to each subset S ⊂ A the binary string y = y1 y2 ...yn where yi = 1 if i ∈ S and 0 otherwise. Such a correspondence is bijective, and the number of ...
Recursive Enumerable
... If HALT is r.e., then Accept(M0) = HALT for some M0 . We’ll want a machine that either accepts or loops forever. So, we define a new machine, M, from M0 as follows. On any input: M behaves just like M0 until M0 stops (if it does). If M0 accepts, then M accepts. If M0 rejects, then M loops forever. ...
... If HALT is r.e., then Accept(M0) = HALT for some M0 . We’ll want a machine that either accepts or loops forever. So, we define a new machine, M, from M0 as follows. On any input: M behaves just like M0 until M0 stops (if it does). If M0 accepts, then M accepts. If M0 rejects, then M loops forever. ...
Lecture 4
... The above properties are easy to prove. Thus, is an equivalence relation on the class of all sets. ...
... The above properties are easy to prove. Thus, is an equivalence relation on the class of all sets. ...
3-8 Unions and Intersection of Sets
... The Union of two or more sets is the set that contains all elements of the sets Symbol: ∪ How to find: list the elements that are in either set, or in both sets ...
... The Union of two or more sets is the set that contains all elements of the sets Symbol: ∪ How to find: list the elements that are in either set, or in both sets ...
on Computability
... rules, can be computed by one of TM. Consistent • Briefly, a Turing machine can be thought of as a black box, which performs a calculation of some kind on an input number. If the calculation reaches a conclusion, or halts then an output number is returned. • One of the consequences of Turing's theor ...
... rules, can be computed by one of TM. Consistent • Briefly, a Turing machine can be thought of as a black box, which performs a calculation of some kind on an input number. If the calculation reaches a conclusion, or halts then an output number is returned. • One of the consequences of Turing's theor ...
(pdf)
... Because of this, it makes sense to restrict our questions about the computability of a function to functions from rational numbers to rational numbers. Because of the bijection between natural numbers and rationals, it also makes sense to further restrict the question to functions from the natural n ...
... Because of this, it makes sense to restrict our questions about the computability of a function to functions from rational numbers to rational numbers. Because of the bijection between natural numbers and rationals, it also makes sense to further restrict the question to functions from the natural n ...
Predicates
... P(n) is computable On input any natural number given in unary describe a TM that decides whether this number is zero (i.e the empty string) or not. • High level program: – If the tape is empty accept else reject ...
... P(n) is computable On input any natural number given in unary describe a TM that decides whether this number is zero (i.e the empty string) or not. • High level program: – If the tape is empty accept else reject ...
