GROUP ACTIONS ON SETS 1. Group Actions Let X be a set and let
... 1. Left multiplication. Let H ≤ G. Then G acts on X = G/H by g(xH) = (gx)H where g, x ∈ G. Note that if xH = yH are the same coset, then (gx)−1 (gy) = x−1 g −1 gy = x−1 y ∈ H, so (gx)H = (gy)H. In other words, this action is well defined. It is also clear that 1(xH) = xH and (gh)(xH) = g(h(xH)), so ...
... 1. Left multiplication. Let H ≤ G. Then G acts on X = G/H by g(xH) = (gx)H where g, x ∈ G. Note that if xH = yH are the same coset, then (gx)−1 (gy) = x−1 g −1 gy = x−1 y ∈ H, so (gx)H = (gy)H. In other words, this action is well defined. It is also clear that 1(xH) = xH and (gh)(xH) = g(h(xH)), so ...
Sets, Numbers, and Logic
... of the unit square was incommensurable with its side: in other words, that 2 was not rational. ( Section A.2 gives his proof.) Suddenly there were a lot of new numbers, irrational numbers. There was no way of representing them except as lengths, that is, as points on a line, a representation not wel ...
... of the unit square was incommensurable with its side: in other words, that 2 was not rational. ( Section A.2 gives his proof.) Suddenly there were a lot of new numbers, irrational numbers. There was no way of representing them except as lengths, that is, as points on a line, a representation not wel ...
Chiron: A Set Theory with Types
... The usefulness of a logic is often measured by its expressivity: the more that can be expressed in the logic, the more useful the logic is. By a logic, we mean a language (or a family of languages) that has a formal syntax and a precise semantics with a notion of logical consequence. (A logic may al ...
... The usefulness of a logic is often measured by its expressivity: the more that can be expressed in the logic, the more useful the logic is. By a logic, we mean a language (or a family of languages) that has a formal syntax and a precise semantics with a notion of logical consequence. (A logic may al ...
On Linear Inference
... may never pick up a given block, even if the rule pickup would permit us to do so. This is more important in this new setting because inferences may be irreversible, so making an inference may constitute a real commitment. If all truths are persistent (and hence inference is monotonic) we can always ...
... may never pick up a given block, even if the rule pickup would permit us to do so. This is more important in this new setting because inferences may be irreversible, so making an inference may constitute a real commitment. If all truths are persistent (and hence inference is monotonic) we can always ...
Supplement: Conditional statements and basic methods of proof
... That is, the truth values of the original conditional statement and its contrapositive agree regardless of the truth values associated with either H or C. This means that the two forms of the statement are logically equivalent in structure. That is, the truth of the contrapositive form ensures the t ...
... That is, the truth values of the original conditional statement and its contrapositive agree regardless of the truth values associated with either H or C. This means that the two forms of the statement are logically equivalent in structure. That is, the truth of the contrapositive form ensures the t ...
Keynote: Structure and Coherence: Telling the Story of the Journey
... apples. How many fewer apples does Lucy have than Julie? ...
... apples. How many fewer apples does Lucy have than Julie? ...
Group knowledge is not always distributed (neither is it always implicit)
... the group is also known to the receiving agent. Note how the axiom A6 and the rule R2 also induce the principle of necessitation on the operator G. The other axioms express that the receiver has the same reasoning and introspection properties as the other agents of the group. In the group knowledge ...
... the group is also known to the receiving agent. Note how the axiom A6 and the rule R2 also induce the principle of necessitation on the operator G. The other axioms express that the receiver has the same reasoning and introspection properties as the other agents of the group. In the group knowledge ...
