# Use the ideas of sets and the definition of the order

Last updated: 7/5/2022

Use the ideas of sets and the definition of the order relation for the whole numbers to justify the transitive property of "less than." That is, if a, b, and c are whole numbers satisfying a <b and b<c, then OA. Since a<b and b<c, choose sets A, B, C satisfying ACBCC and A = {a}, B = {b), and C= {c}. Since each set has the same cardinality and ACC, it follows that a < c. B. Since a<b and b<c, choose sets A, B, C satisfying ACBCC and n(A) = a, n(B) = b, and n(C)= c. Since ACBCC, then aEC. n(C) = c and aEC, so a < c. OC. Since a<b and b<c, choose sets A, B, C satisfying ACBCC and n(A) = a, n(B) = b, and n(C)= c. By the transitive property of set inclusion, ACC, so a < c. OD. Since a<b and b<c, choose sets A, B, C satisfying ACBCC such that A, B, and C are infinite sets with aEA; a and bEB; and a, b, and CEC. By the definition of set inclusion, it follows that a<c.