, , }, C = { , , , }, D = { , , , } E = {– , }, F = { , a }, G = { , – }, H = { , } . Subsets Consider the sets : X = set of all students in your school, Y = set of all students in your class. We note that every element of Y is also an element of X; we say that Y is a subset of X. The fact that Y is subset of X is expressed in symbols as Y ⊂ X.
The symbol ⊂ stands for ‘is a subset of’ or ‘is contained in’. Definition A set A is said to be a subset of a set B if every element of A is also an element of B. In other words, A ⊂ B if whenever a ∈ A, then a ∈ B. It is often convenient to use the symbol “ ⇒ ” which means implies .
Using this symbol, we can write the definiton of subset as follows: A ⊂ B if a ∈ A ⇒ a ∈ B We read the above statement as “ A is a subset of B if a is an element of A implies that a is also an element of B ”. If A is not a subset of B, we write A ⊄ B. We may note that for A to be a subset of B, all that is needed is that every element of A is in B. It is possible that every element of B may or may not be in A.
If it so happens that every element of B is also in A, then we shall also have B ⊂ A. In this case, A and B are the same sets so that we have A ⊂ B and B ⊂ A ⇔ A = B, where “ ⇔ ” is a symbol for two way implications, and is usually read as if and only if (briefly written as “iff”). It follows from the above definition that