For an integer n>4, let S, be the set of all binary strings of length n. The relation R on Sn is defined as the set containing all ordered pairs of binary strings(s, t) such that the first 2 bits of s agree with the first 2 bits of t. Indicate the correct statement.OR is an equivalence relation that partitions S into 4 equivalence classes, each of them of cardinality 2¹-2OR is an equivalence relation that partitions S₂ into 2-2 equivalence classes, each of them of cardinality 4.OR is an equivalence relation that partitions S₂ into 22-4 equivalence classes, each of them of cardinality 16.OR is an equivalence relation that partitions S₂ into 16 equivalence classes, each of them of cardinality 22-4.O None of these.

Given,For an integer , let be the set of all binary strings of length . The relation on is defined as the set containing all ordered pairs of binary strings such that the first bits of agree with the first bits of .Aim: To indicate the correct statement.A relation defined on a set is said to be an equivalence relation, if(i) is reflexive.(ii) is symmetric.(iii) is transitive. The cardinality of a set is the number of elements in a set.The cardinality of the set of all bit-strings of length is . For an integer , let be the set of all binary strings of length .Then a relation on is defined as follows:For all , if and only if the first bits of agree with the first bits of .(i) For all , as all bits are equal. Thus, is reflexive(ii) For all implies that the first bits of agree with the first bits of . This implies that the first bits of agree with the first bits of . Thus, and is symmetric.(iii) Assume that for all . Thus the first bits of agree with the first bits of and the first bits of agree with the first bits of . This further implies that the first bits of agree with the first bits of . Hence and is transitive.Hence, is an equivalence relation.let be the set of all binary strings of length .Since each bit-string is a sequence of binary digits and . Thus, the cardinality of the set ,Let , then the equivalence class of is defined asAny element has the first two bits the same as that of . Thus, there are places left in which can take value or . Hence, the cardinality of each equivalence class is .Since the cardinality of is and the cardinality of each equivalence class is , the number of equivalence classes,Therefore, is an equivalence relation that partitions into 4 equivalence classes, each of them of cardinality..The correct answer is the first option.

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For an integer n>4, let Sn be the set of all binary strings of length n. The relation R on Sn is defined as the set containing all ordered pairs of binary strings (s, t) such that the first 2 bits of s agree with the first 2 bits of t. Indicate the correct statement. OR is an equivalence relation that partitions S₂ into 4 equivalence classes, each of them of cardinality 22-2 OR is an equivalence relation that partitions S₂ into 2"-2 equivalence classes, each of them of cardinality 4. OR is an equivalence relation that partitions S₂ into 22-4 equivalence classes, each of them of cardinality 16. OR is an equivalence relation that partitions S₂ into 16 equivalence classes, each of them of cardinality 22-4. O None of these.

For an integer n>4, let Sn be the set of all binary strings of length n. The relation R on Sn is defined as the set containing all ordered pairs of binary strings (s, t) such that the first 2 bits of s agree with the first 2 bits of t. Indicate the correct statement. OR is an equivalence relation that partitions S₂ into 4 equivalence classes, each of them of cardinality 22-2 OR is an equivalence relation that partitions S₂ into 2"-2 equivalence classes, each of them of cardinality 4. OR is an equivalence relation that partitions S₂ into 22-4 equivalence classes, each of them of cardinality 16. OR is an equivalence relation that partitions S₂ into 16 equivalence classes, each of them of cardinality 22-4. O None of these.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

For an integer n>4, let S, be the set of all binary strings of length n. The relation R on Sn is defined as the set containing all ordered pairs of binary strings
(s, t) such that the first 2 bits of s agree with the first 2 bits of t. Indicate the correct statement.
OR is an equivalence relation that partitions S into 4 equivalence classes, each of them of cardinality 2¹-2
OR is an equivalence relation that partitions S₂ into 2-2 equivalence classes, each of them of cardinality 4.
OR is an equivalence relation that partitions S₂ into 22-4 equivalence classes, each of them of cardinality 16.
OR is an equivalence relation that partitions S₂ into 16 equivalence classes, each of them of cardinality 22-4.
O None of these.

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