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If you have any doubt or need more clarification at any step ease comment .
16. (8 points) Let Z be the integers and let A - Zx Z. Define the relation R on A by (a, b) R(c, d) if and only if a c and b 3 d for all (a, b), (c, d)E A. Prove that R is a partial ordering on A tha...
4. Let S = {1,2,3). Define a relation R on SxS by (a, b)R(c,d) iff a <c and b <d, where is the usual less or equal to on the integers. a. Prove that R is a partial order. Is R a linear order? b. Draw the poset diagram of R.
5. On the set of integers Z define the following relation: "aRb if and only if a - b is a multiple of 7." (1) Prove that R is an equivalence relation. 16 Marks] How many elements are there in the quotient set of 2 with respect to the equivalence relation R? Give reasons. |4 Marks
8.) Consider the integers Z. Dene the relation on Z by x y if and only if 7j(y + 6x). Prove: a.) The relation is an equivalence relation. b.) Find the equivalence class of 0 and prove that it is a subgroup of Z with the usual addition operator on the integers. 8.) Consider the integers Z. Define the relation ~ on Z by x ~ y if and only if 7)(y + 6x). Prove: a.) The relation is an...
5. Let Zli_ {a + bi l a,b E Z. i2--1} be the Gaussian integers. Define a function for all a bi E Zi]. We call N the norm (a) Prove that N is multiplicative. This is, prove that for all a bi, c+di E Z[i] (b) Prove that if a + r є z[i] is a unit of Zli], then Ma + bi)-1. (c) Find all of the units in Zli 5. Let Zli_ {a + bi l a,b...
Discrete Math. Show all steps clearly Define a relation R on the set of all integers Z as follows: Is R a partial order relation? Prove or give a counterexample.
please show the detailed proof,thanks! 1. Tet R be a relation on Zx Z given by (a, b)Rlc,d) if and only if a b or c d. Is R an equivalence relation on Z? Prove your answer.
Discrete Mathematics 22. Let r be a relation on the integers such that (a, b) E r if and only if a +b 1. What is the transitive closure of r? 23. Write an algorithm in pseudo code that converts numbers in decimal representation to octal (base 8) representation 24. Prove that the set of integers in countable 22. Let r be a relation on the integers such that (a, b) E r if and only if a +b 1....
(e) Define a relation R on Z as xRy if and only if m|(x - y). Prove that R is an equiv- alence relation.
4. Let 3 be the relation on Z2 defined by (a,b) 3 (c,d) if and only if a Sc and b < d. (a) Prove that is a partial order. (b) Find the greatest lower bound of {(1,5), (3,3)}. (c) Is < a total order? Justify your answer.
Let R be the relation defined on Z (integers): a R b iff a + b is even. Then the distinct equivalence classes are: Group of answer choices [1] = multiples of 3 [2] = multiples of 4 [0] = even integers and [1] = the odd integers all the integers None of the above