(6) Prove that for all xeR, x > 0 there exists n eN such that 름 <z. (7) Prove that is irrational. (One or both numbers will be different, of course.) (6) Prove that for all xeR, x > 0 there exists n eN such that 름
Prove that the following relation R is an equivalence relation on the set of ordered pairs of real numbers. Describe the equivalence classes of R. (x, y)R(w, z) y-x2 = z-w2
4. Prove that isomorphism is an equivalence relation on the set of all rings. You may assume elementary results regarding the inverses and compositions of bijections. Proof:
Is the exclusive disjunction (operator ) of two equivalence relations also an equivalence relation? Prove your conclusion formally. Subject Is the exclusive disjunction (operator ) of two equivalence relations also an equivalence relation? Prove your conclusion formally.
2. Find all square-free perfect numbers. (Prove that there are no others.) 2. Find all square-free perfect numbers. (Prove that there are no others.)
Please do problem 9 and write a detailed proof when doing (a) 9. Letbe the relation on the set of non-zero real numbers defined as follows: for r, y E R [0), x~ylf and only if-EQ (a) Prove thatis an equivalence relation. (b) Determine the equivalence class of π. 9. Letbe the relation on the set of non-zero real numbers defined as follows: for r, y E R [0), x~ylf and only if-EQ (a) Prove thatis an equivalence relation. (b)...
2. Consider the relation E on Z defined by E n, m) n+ m is even} equivalence relation (a) Prove that E is an (b) Let n E Z. Find [n]. equivalence relation in [N, the equivalence class of 3. We defined a relation on sets A B. Prove that this relation is an (In this view, countable sets the natural numbers under this equivalence relation). exactly those that are are 2. Consider the relation E on Z defined by...
4. Prove that {(x,y) e R2 : x - ye Q} is an equivalence relation on the set of re denotes the set of rational numbers
9. Let R an equivalence relation. Prove or disprove that R:R is an equivalence relation
4. Convert the following regular expressions to e-NFA's. (a) 1(0110)0(11 10) (b) (000)(011+001) (111) (c) (01 10(00 11)(1 10 100)