ds-t3-a

Prove that if r is a reflexive relation on a nonempty

Info iconThis preview shows pages 3–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Prove that if R is a reflexive relation on a nonempty set A, then R R n for every positive integer n. // Remark: Evidently we need to do a proof by induction since the n th composition power of a relation is defined recursively. Proof. The basis step is true since R 1 = R implies R R 1 for any relation R on a nonempty set A. To deal with the induction step, it suffices to show that if the relation R is reflexive, then ( n ε + )( R R n R R n+1 ). To this end, pretend n is an arbitrary positive integer, R is reflexive, and R R n . Let (a,b) ε R be arbitrary. Then (a,b) ε R and the induction hypothesis, R R n , implies (a,b) ε R n . R reflexive and a ε A (a,a) ε R. The definition of the composition R n R=R n+1 , (a,a) ε R, and (a,b) ε R n implies (a,b) ε R n+1 . Thus we have shown that R R n R R n+1 for an arbitrary n ε + . So we have verified -- ugh -- that ( n ε + )( R R n R R n+1 ). Since we have verified the hypotheses of the induction axiom, modus ponens wraps things up: If R is reflexive, ( n ε + )( R R n ).//
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
TEST3/MAD2104 Page 4 of 4 _________________________________________________________________ 7. (15 pts.) (a) How many vertices does a tree with 37 edges have? V= E + 1 = 3 8 (b) What is the maximum number of leaves that a binary tree of height 6 can have? If l denotes the number of leaves, then l 2 6 = 64. The maximum occurs when the tree is complete, i.e., all the leaves are at the same level. (c) If a full 3-ary tree has 24 internal vertices, how many leaves does it have?
Background image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page3 / 4

Prove that if R is a reflexive relation on a nonempty set A...

This preview shows document pages 3 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online