Discrete Mathematics with Graph Theory (3rd Edition) 123

# Discrete Mathematics with Graph Theory (3rd Edition) 123 -...

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Section 5.1 121 the induction hypothesis). Since x rt A k+1, x belongs to an odd number of the sets Al, A 2, ... ,Ak+1. On the other hand, if x E A k + 1 " S, then, since x rt s, we know that x belongs to an even number (possibly zero) of the sets A 1, A 2, . .. ,Ak, again by the induction hypothesis. Since x E Ak+1' we conclude again that x belongs to an odd number of the sets A 1, A 2, . .. , Ak+1. This proves that the symmetric difference of A 1 , .•. , A k + 1 consists of elements which belong to an odd number of these sets. Conversely, suppose x belongs to an odd number of the sets A 1, ... , Ak+1. We must show x belongs to the symmetric difference of A 1, ... , Ak+1. If x belongs to A k+1, then x belongs to an even number of the sets A 1, . .. , Ak and, hence, x E Ak+1 " S by the induction hypothesis. However, if x does not belong to Ak+lo then clearly xES" A k+1. Thus, x belongs to the symmetric difference of Al, . .. ,Ak+1. By the Principle of Mathematical Induction, we conclude that for all
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