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Assignment 7 - Assignment#7 Section 3.1 11 13c Drawings...

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Assignment #7 Section 3.1 11) Drawings. 13c) Rng(S o R) is a subset of Rng(S) Take the domain to be the real numbers. Define S(x)=x^3 and R(x)=x^2 for x a real number. Then Rng(S) is all of the reals, but Rng(S o R) includes only the positive real. S o R may put a restriction on the domain of S so the Rng(S o R) is a subset of Rng(S) Section 3.2 1a) {(1,2)} on A={1,2} Reflexivity: Take 1, (1,1) is not in the relation ==> Not reflexive Symmetry: Take (1,2) in the relation. (2,1) is not in the relation ==> Not Symmetric Transitivity: (1,2) is in the relation but there is no (2,x) in the relation, for some x in A. So, antecedent is false. ==> Transitive 1c) = on N Suppose a, b, c are natural numbers. Reflexivity: If a is a natural number, a = a. ==>Reflexive. Symmetry: If a = b, then b = a. ==> Symmetric. Transitivity: If a = b, and b = c, then a = c. ==> Transitive. 1d) < on N Suppose a, b, c are natural numbers. Reflexivity: If a is a natural number, a is not less than a. ==> Not Reflexive. Symmetry: If a < b then b > a ==> Not Symmetric. Transitivity: If a < b and b < c , then a < c. ==> Transitive. 1f) <> on N Suppose a, b, c are natural numbers. Reflexivity: If a is a natural number, a = a. ===> Not Reflexive. Symmetry: If a <> b, then b <> a. ==> Symmetric. Transitiviy: If a <> b, and b <> c, it does not necessarily mean that a <> c. eg. 3 <> 4 and 4 <> 3, but 3 = 3. ==> Not Transitive.
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1j) R, where (x,y) R (z,w) iff x + z =< y + w on the set R x R Reflexivity: example: (2,1) is not related to itself. 2 + 2 > 1 + 1. ===> Not Reflexive. Symmetry: If (a,b) R (c,d), then a + c =< b + d iff c + a =< d + b so (c,d) R (a,b) ===> Symmetric. Transitivity: example: (1,0) R (2,4), (2,4) R (2,1), but (1,0) not related to (2,1) ===> Not Transitive. 2a) (1,3), (2,3) (different answers are possible) 2b) (1,1), (2,2), (3,3), (1,3), (3,2) (there are different answers possible) 2c) (1,2), (2,1), (1,3), (3,1) a R b iff the prime factorizations of a and b have same # of 2's.
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