Sets And Relations

To find the inverse of 2 with respect to the binary operation * (defined by ), we need an such that , where is the identity element for the operation.First, let's determine the identity element : Thus, the identity element is 3.Next, we need to find such that: Therefore, the inverse of 2 with respect to the binary operation * is 4.

$ R X \forall A \in X R A A=A A A \in X (A, A) \in R A \in X R R X A, B \in X A B=B A B A=A B (A, B) \in R (B, A) \in R R R X A, B, C \in X (A, B) \in R (B, C) \in R \Rightarrow(A, C) \in R A B=B A B C=C B A C=C A 2 \times 2 A B=B A B C=C B A C \neq C A R$ is not transitive.

To determine the properties of the relation on the set of all matrices with real entries defined by we need to check if is reflexive, symmetric, and transitive.Reflexivity:A relation is reflexive if every element is related to itself, i.e., for all .Given any matrix , we have .Since is an integer, for all . Therefore, is reflexive.Symmetry:A relation is symmetric if implies .Suppose , meaning is an integer. We need to verify that is also an integer.Since and the negative of an integer is also an integer, is an integer. Thus, is symmetric.Transitivity:A relation is transitive if whenever and , it follows that .Let's consider: We find: Here, and , but because , which is not an integer. Hence, is not transitive.Based on this analysis, is reflexive and symmetric, but not transitive.