examples which are of great importance for various branches of mathematics, like com-pact Lie groups, Grassmannians and bounded symmetric domains. A bijective function composed with its inverse, however, is equal to the identity. Relations ≥ and = on the set N of natural numbers are examples of weak order, as are relations ⊇ and = on subsets of any set. R is re exive if, and only if, 8x 2A;xRx. Proof. Recall: 1. • The linear model assumes that the relations between two variables can be summarized by a straight line. Example 2.4.1. For example, Q i are linear orders. De nition 3. I Symmetric functions are closely related to representations of symmetric and general linear groups Kernel Relations Example: Let x~y iff x mod n = y mod n, over any set of integers. R is transitive x R y and y R z implies x R z, for all x,y,z∈A Example: i<7 and 7 and … are examples of strict orders on the corresponding sets. I Some combinatorial problems have symmetric function generating functions. • Correlation means the co-relation, or the degree to which two variables go together, or technically, how those two variables covary. Properties of real symmetric matrices I Recall that a matrix A 2Rn n is symmetric if AT = A. I For real symmetric matrices we have the following two crucial properties: I All eigenvalues of a real symmetric matrix are real. REMARK 25. 2. • Measure of the strength of an association between 2 scores. 2 are equivalence relations on a set A. Two elements a and b that are related by an equivalence relation are called equivalent. Problem 2. Give the rst two steps of the proof that R is an equivalence relation by showing that R is re exive and symmetric. I Symmetric functions are useful in counting plane partitions. The parity relation is an equivalence relation. 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