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x = ABCDE y = ABCDE (x == y) = false x.equals(y) = true Here, the two objects x and y (or, more precisely, the two objects that are referenced by the reference variables x and y) are different objects, occupying different memory locations, so they are not identically equal: (x == y) evaluates to false at line 7. But they do both have the same contents, so they are mathematically equal, and x.equals(y) evaluates to true at line 8.

The distinction between identical equality and mathematical equality exists in Java only for reference variables (i.e., only for objects). For all variables of primitive types, the equality operator tests for mathematical equality.

in a given month to the prices that existed during a reference period. The producer price index (PPI) and GDP de ator are the other two major price indexes. The PPI measures the prices for nished goods, intermediate materials, and crude materials at the wholesale level. Because wholesale prices are eventually translated into retail prices, changes in the PPI are usually a good predictor of changes in the CPI. The GDP de ator is the most comprehensive measure of the price level since it measures prices for net exports, investment, and government expenditures, as well as for consumer spending. In ation is the annual rate of increase in the price level. Disin ation is a term used to denote a slowdown in the rate of in ation; de ation exists when there is an annual rate of decrease in the price level. While there have been some monthly decreases in the price level, the U.S. economy has not experienced de ation since the 1930s.

Data structures have both content and structure. So it is possible for two data structures to have equal contents (i.e., have the same contents) but be organized differently. For example, two arrays could both contain the three numbers 22, 44, and 88, but in different orders. EXAMPLE 11.7 Testing Equality of Arrays

1 2 3 4 5 6 7 8 9 10 11 12 13 14

public class TestArraysEquality { public static void main(String[] args) { int[] x = { 22, 44, 88 }; int[] y = { 88, 44, 22 }; ch02.ex02.DuplicatingArrays.print(x); ch02.ex02.DuplicatingArrays.print(y); System.out.println("Arrays.equals(x, y) = " + Arrays.equals(x, y)); Arrays.sort(x); Arrays.sort(y); ch02.ex02.DuplicatingArrays.print(x); ch02.ex02.DuplicatingArrays.print(y); System.out.println("Arrays.equals(x, y) = " + Arrays.equals(x, y)); } }

The output it:

{22, 44, 88} {88, 44, 22} Arrays.equals(x, y) = false {22, 44, 88} {22, 44, 88} Arrays.equals(x, y) = true This shows that the java.util.Arrays.equal() method requires not only the same contents for

In ation refers to an increase in the general price level, not the price of a speci c good or service.

Equality is a weaker relation than identity: Identical objects are always equal, but equal objects may not be identical; they could be distinct. Equality of data structures means the same structure and the same contents in the same order. A weaker kind of reflexive relation is isomorphism. Two data structures are isomorphic if they have the same structure. This concept is used when the data part of the data structure is irrelevant. Two arrays are isomorphic if they have the same length. Two trees are isomorphic if one tree can be rearranged to match the other. More formally, T1 is isomorphic to T2 (sometimes written T1 T2 ) if there is a one-to-one mapping (an isomorphism) between them that preserves parent-child relationship between all nodes. EXAMPLE 11.8 Isomorphic Trees

As unordered trees, Tree 1 and Tree 2 in Figure 11.7 are isomorphic, but not equal. However, Tree 3 is not isomorphic to either of the other two trees because it has only three leaves; the other two trees each have four leaves:Tthat s a different structure. That distinction leads fairly easily to a formal deduction that there is no isomorphism between Tree 1 and Tree 3. As ordered trees, Tree 1 is not isomorphic to Tree 2 because their roots left-most subtrees have different sizes. The left-most subtree in Tree 1 has three nodes, while that of Tree 2 has only two nodes. That distinction also leads fairly easily to a formal deduction that no isomorphism between Tree 1 and Tree 2 can exist.

CHAP. 11]