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proper subset symbol

The proper subset symbol indicates a specific relationship between two set s. The symbol looks like the uppercase letter U in a sans-serif font , rotated 90 degrees clockwise. Proper-subset relations form the foundation of mathematical logic, including Boolean algebra, which is important in the design of computers and signal-processing systems.

Suppose there are two sets A and B. The statement "Set A is a proper subset of set B" is written A B. This means that every element contained in set A is also contained in set B, and there is at least one element in B that is not contained in A. Thus, no set is a proper subset of itself.

Here are some examples of true statements using the proper subset symbol:

{1, 2, 3, 4, 5, ... } {0, 1, 2, 3, 4, ...}
{0, 1, 2, 3, 4} {0, 1, 2, 3, 4, ...}
{-2, -3, 4} {-2,-2.5, -3, -3.5, -4}

The following statements, however, are not true:

{0, 1, 2, 3, 4, ...} {0, 1, 2, 3, 4, ...}
{0, 1, 2, 3, 4, ...} {0, -1, -2, -3, -4, ...}
{0, 1, 2, 3, 4, ...} {0, 1, 2, 3, 4}

Sets can contain things other than numbers. Examples are points on a plane, points on a spherical surface, and points in three-dimensional (3D) space. Proper subset relationships can be expressed in terms of specialized illustrations called Venn diagram s. In the illustration below, A B, and C B. However, it is not true that B A, nor is it true that A C, nor is it true that B C. In addition, the following statements are false: A A, B B, and C C.

Compare subset symbol . Also see set theory and Mathematical Symbols .

This was last updated in September 2005

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