lecture notes 5
... Theorem 2.5. Let G, H be groups and let f : G → H be a group homomorphism. Then the kernel of f is a normal subgroup of G, and the quotient group G/ ker f is isomorphic to the image of f . Proof. Clearly, ker f contains the unit element of G, by the definition of a group homomorphism (it must send 1 ...
... Theorem 2.5. Let G, H be groups and let f : G → H be a group homomorphism. Then the kernel of f is a normal subgroup of G, and the quotient group G/ ker f is isomorphic to the image of f . Proof. Clearly, ker f contains the unit element of G, by the definition of a group homomorphism (it must send 1 ...
Simplifying Algebraic Expressions Involving Exponents
... The ratio of the surface area to the volume of microorganisms affects their ability to survive. An organism with a higher surface area-to-volume ratio is more buoyant and uses less of its own energy to remain near the surface of a liquid, ...
... The ratio of the surface area to the volume of microorganisms affects their ability to survive. An organism with a higher surface area-to-volume ratio is more buoyant and uses less of its own energy to remain near the surface of a liquid, ...
Section 3.6: Indirect Argument: Contradiction and Contraposition
... So far, we have only considered so called “direct proofs” of mathematical statements. Specifically, we have been given a statement to prove, and then we have used the definitions and previous results to logically derive the statement. In this section we consider “indirect proofs” proofs which do not ...
... So far, we have only considered so called “direct proofs” of mathematical statements. Specifically, we have been given a statement to prove, and then we have used the definitions and previous results to logically derive the statement. In this section we consider “indirect proofs” proofs which do not ...
A Complexity of Two-variable Logic on Finite Trees
... In particular, we will consider: — satisfiability in the presence of all navigational predicates: predicates for the parent/child relation, its transitive closure (the descendant relation), the previous/next sibling relation and its transitive closure; — satisfiability when the signature is restrict ...
... In particular, we will consider: — satisfiability in the presence of all navigational predicates: predicates for the parent/child relation, its transitive closure (the descendant relation), the previous/next sibling relation and its transitive closure; — satisfiability when the signature is restrict ...
