Nt1310 Unit 4 Lab Report

1. A) Show that the relation R over bit strings where (x, y) is in R if and only bit strings x and y length 16 that agree on their last 4 bits is an equivalence relation. Define the equivalence classes and the partition induced by R.
Answer:
A relation R is said to be an equivalence relation if and only if it has all the following three properties:
• Reflexive
• Symmetric and
• Transitive

We got to show that the relation R over bit strings where (x, y) is in R if and only bit strings x and y length 16 that agree on their last 4 bits is an equivalence relation. To prove this we got to prove the relation R is following all the 3 properties as mentioned before.
Let us consider x’s equivalent bit string of length 16 is x0x1x2x3x4x5x6x7x8x9x10x11x¬12x13x14x15

Let set S2 take all the values from the domain of relation R where the 13th bit of the equivalent bit string is 1.
i.e., Set S2 contains all the values whose last four bits are of the form 1x13x14x15 where last three bits can take either 0 or 1 value.

P2

Consider Partition P1 has three sets S3, S4 and S5
Let set S3 take all the values from the domain of relation R where the 13th bit of the equivalent bit string is 0.
i.e., Set S3 contains all the values whose last four bits are of the form 0x13x14x15 where last three bits can take either 0 or 1 value.

Let set S4 take all the values from the domain of relation R where 13th bit is 1 and the 14th bit is 0 of the equivalent bit string.
i.e., Set S4 contains all the values whose last four bits are of the form 10x14x15 where last two bits can take either 0 or 1 value.
Let set S5 take all the values from the domain of relation R where 13th bit is 1 and the 14th bit is 1 of the equivalent bit string.
i.e., Set S5 contains all the values whose last four bits are of the form 11x14x15 where last two bits can take either 0 or 1