denoted by , in analogue with the notation for binomial coefficients, which count the number of -subsets of an -element set. In set theory, the notation is also common, especially when is a transfinite cardinal number. Properties A set A is a subset of B if and only if their intersection is equal to A. Formally: A set A is a subset of B if and only if their union is equal to B. Formally: A finite set A is a subset of B, if and only if the cardinality of their intersection is equal to the cardinality of A. Formally: ⊂ and ⊃ symbols Some authors use the symbols and to indicate and respectively; that is, with the same meaning as and instead of the symbols and For example, for these authors, it is true of every set A that Other authors prefer to use the symbols and to indicate (also called strict) subset and superset respectively; that is, with the same meaning as and instead of the symbols and This usage makes and analogous to the inequality symbols and For example, if then x may or may not equal y, but if then x definitely does not equal y, and is less than y. Similarly, using the convention that is proper subset, if then A may or may not equal B, but if then A definitely does not equal B. Examples of subsets The set A = {1, 2} is a proper subset of B = {1, 2, 3}, thus both expressions and are true. The set D = {1, 2, 3} is a subset (but a proper subset) of E = {1, 2, 3}, thus is true, and is not true (false). Any set is a subset of itself, but not a proper subset. ( is true, and is false for any set X.) The set {x: x is a prime number greater than 10} is a proper subset of {x: x is an odd number greater than 10} The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a line segment is a proper subset of the set of points in a line. These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number