Label each of the following statements as either true or false.
Every mapping on a nonempty set
Whether the statement, “Every mapping on the nonempty set
Answer to Problem 1TFE
Solution:
The statement, “Every mapping on the nonempty set
Explanation of Solution
Consider the statement, “Every mapping on the nonempty set
Let
A relation on a nonempty set
From the above definitions,
Every mapping is a relation.
Hence the statement, “Every mapping on the nonempty set
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Chapter 1 Solutions
Elements Of Modern Algebra
- True or False Label each of the following statements as either true or false. 2. Every relation on a nonempty set is as mapping.arrow_forwardGive an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.arrow_forwardTrue or False Label each of the following statements as either true or false. Let be an equivalence relation on a nonempty setand let and be in. If, then.arrow_forward
- Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.arrow_forwardLabel each of the following statements as either true or false. If R is an equivalence relation on a nonempty set A, then any two equivalence classes of R contain the same number of element.arrow_forwardLabel each of the following statements as either true or false. 9. Composition of mappings is an associative operation.arrow_forward
- True or False Label each of the following statements as either true or false. If is an equivalence relation on a nonempty set, then the distinct equivalence classes of form a partition of.arrow_forwardLabel each of the following statements as either true or false. 3. Let , , and be mappings from into such that . Then .arrow_forwardLet (A) be the power set of the nonempty set A, and let C denote a fixed subset of A. Define R on (A) by xRy if and only if xC=yC. Prove that R is an equivalence relation on (A).arrow_forward
- In each of the following parts, a relation is defined on the set of all human beings. Determine whether the relation is reflective, symmetric, or transitive. Justify your answers. xRy if and only if x lives within 400 miles of y. xRy if and only if x is the father of y. xRy if and only if x is a first cousin of y. xRy if and only if x and y were born in the same year. xRy if and only if x and y have the same mother. xRy if and only if x and y have the same hair colour.arrow_forwardSuppose f,g and h are all mappings of a set A into itself. a. Prove that if g is onto and fg=hg, then f=h. b. Prove that if f is one-to-one and fg=fh, then g=h.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,