Why Semigroups Are Important?

Pliz help me, I have Set theory, Relations and Groups Problems?

• A1. Determine whether the relation xRy where xRy means there exists an integer n such that x=2n y is an equivalence relation. Is the relation antisymmetric? A2. Define an operation * on R by x*y=xy +1. Show that * is commutative but not associative. If we consider R\{0) does * have an identity on this set? A3. Let a,b be fixed rel numbers and define θ:R→R by θ(x)=ax+b. Prove that θ is bijective if and only if a≠0 and give a formula for θ-1. A4. Prove that (AxB)∩(BxA)=(A∩B)x(A∩B) for any sets A,B. Is the result true if we replace ∩by Ù? A5. Let n be an integer. Prove that if n2 is even then n is even. Hint: Assume that n is not even, ie. N is odd. A6. Determine whether the relation “x divides y” is a total order on the set W={1,2,4}. A7. Let M denote the set of all invertible 2x2 matrices with real entries. Prove that M is a commutative group under addition. A8. Let S =P(A) (the set of all subsets of A). Do the semigroups (S, V ) and (S,^) have identities? A9. Let (A,*) be a set closed under * with identity 1 and a,x be elements of A with inverses b and y respectively. What is the inverse of a*x in A? A10. Let f:R→R and g:R→R be functions defined by f(x)= 2x+1 and g(x)+x2-2. Determine which one is injective and which one is surjective. Determine whether f and g are bijective. Determine (g о f)(2). A11. Determine whether f:R→R given by f(x)=√x is a mapping.

• Answer:

  i will help you with some.. . . 2 .. . . x*y = xy + 1 while y*x = yx + 1 = xy + 1 ... thus the relation is commutative now, consider (1 * (2 * 3)) = 1 * 7 = 8 then ((1 * 2) * 3) = (3 * 3) = 10 thus it is not associative. 3. θ^(-1) = (x - b) / a ... thus it is necessary that a ≠ 0 5. supposing n is odd, then n = (2k+1) for some integer k then n^2 = 4k^2 + 4k + 1 = odd thus we contradict the hypothesis. thus if n is odd, then n^2 is odd... by contrapositivity if n^2 is even, then n is even. (it also needs law of excluded middle, n is not odd means n is even) .. . .. .