Indicate whether the following statements are true or false. For all equivalence relations on the set X, and a, b E X, if ab, then be [a]. For all sets X and partitions P of X, (X/~p) OP= P. The set {x ER | 1 < x <2} is countable. For all sets A, B, and C, if A has the same cardinality as B and B has the same cardinality as C, then A has the same cardinality as C. The set of negative rational numbers is countable.

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A: A relation R is said to be an equivalence relation, if it is  Reflexive : For every a∈R→(a,a)∈R…

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Q: In the Axiom Systems section of the Background chapter (1.6), a binary relation <on a nonempty set X…

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Q: Let A be a set and R and S be relations on A. Define the relation R+S on A by R+S = (R – S)U(S – R).…

A: Given A be a set and R and S be relations on A. R+S on A is defined by R+S=R-S∪S-R=R∪S-R∩S=R∪S∩R∩Sc…

Q: Let (A= (2, 4, 6, 9, 12, 18, 27, 36, 48, 60, 72), R) be a partially ordered set, where the relation…

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Q: Another example of a relation is the divisibility relation (denoted by ), that is, for A=N, where N…

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Q: 1. Verify that the relation ~ is an equivalence relation on the set S given. (a) S = R reals, a ~ b…

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Q: Given set A = {1, 2, 3} and S = P(A) The relation is defined as xRy ⇔ |x| = |y| Indicate the…

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Q: Define a binary relation on the set S=(-3, -2, -1, 1, 2, 3} as follows: R= {(a,b) ab>0} %3D a. Find…

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Q: Give an example of two equivalence relations R and S on the set A = {1, 2, 3} such that RUS is not…

A: 6 Given set is A=1,2,3. Consider the relation R and S on A is given by  R=1,1,2,2,3,31,2,2,1 And…

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Q: Let A = {a, b, c, d, e}. Suppose R is an equivalence relation on A. Suppose R has two equivalence…

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Q: Suppose A and B are two finite sets such that |B| = 2 |A| and the number of relations on A is 16.…

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Q: Consider the set S = {A, B, C, D}: For each of the following relations on S, determine whether it is…

A: Definitions: