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Preliminaries: exercises
Preliminaries: exercises — from Judson, Abstract Algebra: Theory and Applications.
Practice (29)
Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.
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Suppose that \[\begin{aligned}A & = \{ x : x \in \mathbb N \text{ and } x \text{ is even} \}, \\ B & = \{x : x \in \mathbb N \text{ and } x \text{ is prime}\}, \\ C & = \{ x : x \in \mathbb N \text{ and } x \text{ is a multiple of } 5\}\end{aligned}\]. Describe each of the following sets.
\(A \cap B\)
\(B \cap C\)
\(A \cup B\)
\(A \cap (B \cup C)\)
መልሱን አሳይ
Hint:
(a) \(A \cap B = \{ 2 \}\); (b) \(B \cap C = \{ 5 \}\).
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If \(A = \{ a, b, c \}\), \(B = \{ 1, 2, 3 \}\), \(C = \{ x \}\), and \(D = \emptyset\), list all of the elements in each of the following sets.
\(A \times B\)
\(B \times A\)
\(A \times B \times C\)
\(A \times D\)
መልሱን አሳይ
Hint:
(a) \(A \times B = \{ (a,1), (a,2), (a,3), (b,1), (b,2), (b,3), (c,1), (c,2), (c,3) \}\); (d) \(A \times D = \emptyset\).
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Find an example of two nonempty sets \(A\) and \(B\) for which \(A \times B = B \times A\) is true.
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Prove \(A \cup \emptyset = A\) and \(A \cap \emptyset = \emptyset\).
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Prove \(A \cup B = B \cup A\) and \(A \cap B = B \cap A\).
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Prove \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\).
መልሱን አሳይ
Hint:
Observe that \(x \in A \cup B\) if and only if \(x \in A\) or \(x \in B\). Equivalently, \(x \in B\) or \(x \in A\), which is the same as \(x \in B \cup A\). Therefore, \(A \cup B = B \cup A\).
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Prove \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\).
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Prove \(A \subset B\) if and only if \(A \cap B = A\).
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Prove \((A \cap B)' = A' \cup B'\).
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Prove \(A \cup B = (A \cap B) \cup (A \setminus B) \cup (B \setminus A)\).
መልሱን አሳይ
Hint:
\((A \cap B) \cup (A \setminus B) \cup (B \setminus A) = (A \cap B) \cup (A \cap B') \cup (B \cap A') = [A \cap (B \cup B')] \cup (B \cap A') = A \cup (B \cap A') = (A \cup B) \cap (A \cup A') = A \cup B\).
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Prove \((A \cup B) \times C = (A \times C ) \cup (B \times C)\).
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Prove \((A \cap B) \setminus B = \emptyset\).
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Prove \((A \cup B) \setminus B = A \setminus B\).
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Prove \(A \setminus (B \cup C) = (A \setminus B) \cap (A \setminus C)\).
መልሱን አሳይ
Hint:
\(A \setminus (B \cup C) = A \cap (B \cup C)' = (A \cap A) \cap (B' \cap C') = (A \cap B') \cap (A \cap C') = (A \setminus B) \cap (A \setminus C)\).
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Prove \(A \cap (B \setminus C) = (A \cap B) \setminus (A \cap C)\).
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Prove \((A \setminus B) \cup (B \setminus A) = (A \cup B) \setminus (A \cap B)\).
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Which of the following relations \(f: {\mathbb Q} \rightarrow {\mathbb Q}\) define a mapping? In each case, supply a reason why \(f\) is or is not a mapping.
\(\displaystyle f(p/q) = \frac{p+ 1}{p - 2}\)
\(\displaystyle f(p/q) = \frac{3p}{3q}\)
\(\displaystyle f(p/q) = \frac{p+q}{q^2}\)
\(\displaystyle f(p/q) = \frac{3 p^2}{7 q^2} - \frac{p}{q}\)
መልሱን አሳይ
Hint:
(a) Not a map since \(f(2/3)\) is undefined; (b) this is a map; (c) not a map, since \(f(1/2) = 3/4\) but \(f(2/4)=3/8\); (d) this is a map.
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Determine which of the following functions are one-to-one and which are onto. If the function is not onto, determine its range.
\(f: {\mathbb R} \rightarrow {\mathbb R}\) defined by \(f(x) = e^x\)
\(f: {\mathbb Z} \rightarrow {\mathbb Z}\) defined by \(f(n) = n^2 + 3\)
\(f: {\mathbb R} \rightarrow {\mathbb R}\) defined by \(f(x) = \sin x\)
\(f: {\mathbb Z} \rightarrow {\mathbb Z}\) defined by \(f(x) = x^2\)
መልሱን አሳይ
Hint:
(a) \(f\) is one-to-one but not onto. \(f({\mathbb R} ) = \{ x \in {\mathbb R} : x \gt 0 \}\). (c) \(f\) is neither one-to-one nor onto. \(f(\mathbb R) = \{ x : -1 \leq x \leq 1 \}\).
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Let \(f :A \rightarrow B\) and \(g : B \rightarrow C\) be invertible mappings; that is, mappings such that \(f^{-1}\) and \(g^{-1}\) exist. Show that \((g \circ f)^{-1} =f^{-1} \circ g^{-1}\).
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Define a function \(f: {\mathbb N} \rightarrow {\mathbb N}\) that is one-to-one but not onto.
Define a function \(f: {\mathbb N} \rightarrow {\mathbb N}\) that is onto but not one-to-one.
መልሱን አሳይ
Hint:
(a) \(f(n) = n + 1\).
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Prove the relation defined on \({\mathbb R}^2\) by \((x_1, y_1 ) \sim (x_2, y_2)\) if \(x_1^2 + y_1^2 = x_2^2 + y_2^2\) is an equivalence relation.
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Let \(f : A \rightarrow B\) and \(g : B \rightarrow C\) be maps.
If \(f\) and \(g\) are both one-to-one functions, show that \(g \circ f\) is one-to-one.
If \(g \circ f\) is onto, show that \(g\) is onto.
If \(g \circ f\) is one-to-one, show that \(f\) is one-to-one.
If \(g \circ f\) is one-to-one and \(f\) is onto, show that \(g\) is one-to-one.
If \(g \circ f\) is onto and \(g\) is one-to-one, show that \(f\) is onto.
መልሱን አሳይ
Hint:
(a) Let \(x, y \in A\). Then \(g(f(x)) = (g \circ f)(x) = (g \circ f)(y) = g(f(y))\). Thus, \(f(x) = f(y)\) and \(x = y\), so \(g \circ f\) is one-to-one. (b) Let \(c \in C\), then \(c = (g \circ f)(x) = g(f(x))\) for some \(x \in A\). Since \(f(x) \in B\), \(g\) is onto.
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Define a function on the real numbers by \[\begin{aligned}\end{aligned}\]. What are the domain and range of \(f\)? What is the inverse of \(f\)? Compute \(f \circ f^{-1}\) and \(f^{-1} \circ f\).
መልሱን አሳይ
Hint:
\(f^{-1}(x) = (x+1)/(x-1)\).
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Let \(f: X \rightarrow Y\) be a map with \(A_1, A_2 \subset X\) and \(B_1, B_2 \subset Y\).
Prove \(f( A_1 \cup A_2 ) = f( A_1) \cup f( A_2 )\).
Prove \(f( A_1 \cap A_2 ) \subset f( A_1) \cap f( A_2 )\). Give an example in which equality fails.
Prove \(f^{-1}( B_1 \cup B_2 ) = f^{-1}( B_1) \cup f^{-1}(B_2 )\), where \[\begin{aligned}\end{aligned}\].
Prove \(f^{-1}( B_1 \cap B_2 ) = f^{-1}( B_1) \cap f^{-1}( B_2 )\).
Prove \(f^{-1}( Y \setminus B_1 ) = X \setminus f^{-1}( B_1)\).
መልሱን አሳይ
Hint:
(a) Let \(y \in f(A_1 \cup A_2)\). Then there exists an \(x \in A_1 \cup A_2\) such that \(f(x) = y\). Hence, \(y \in f(A_1)\) or \(f(A_2)\). Therefore, \(y \in f(A_1) \cup f(A_2)\). Consequently, \(f(A_1 \cup A_2) \subset f(A_1) \cup f(A_2)\). Conversely, if \(y \in f(A_1) \cup f(A_2)\), then \(y \in f(A_1)\) or \(f(A_2)\). Hence, there exists an \(x\) in \(A_1\) or \(A_2\) such that \(f(x) = y\). Thus, there exists an \(x \in A_1 \cup A_2\) such that \(f(x) = y\). Therefore, \(f(A_1) \cup f(A_2) \subset f(A_1 \cup A_2)\), and \(f(A_1 \cup A_2) = f(A_1) \cup f(A_2)\).
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Determine whether or not the following relations are equivalence relations on the given set. If the relation is an equivalence relation, describe the partition given by it. If the relation is not an equivalence relation, state why it fails to be one.
\(x \sim y\) in \({\mathbb R}\) if \(x \geq y\)
\(m \sim n\) in \({\mathbb Z}\) if \(mn > 0\)
\(x \sim y\) in \({\mathbb R}\) if \(|x - y| \leq 4\)
\(m \sim n\) in \({\mathbb Z}\) if \(m \equiv n \pmod{6}\)
መልሱን አሳይ
Hint:
(a) The relation fails to be symmetric. (b) The relation is not reflexive, since \(0\) is not equivalent to itself. (c) The relation is not transitive.
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Define a relation \(\sim\) on \({\mathbb R}^2\) by stating that \((a, b) \sim (c, d)\) if and only if \(a^2 + b^2 \leq c^2 + d^2\). Show that \(\sim\) is reflexive and transitive but not symmetric.
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Show that an \(m \times n\) matrix gives rise to a well-defined map from \({\mathbb R}^n\) to \({\mathbb R}^m\).
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Find the error in the following argument by providing a counterexample. The reflexive property is redundant in the axioms for an equivalence relation. If \(x \sim y\), then \(y \sim x\) by the symmetric property. Using the transitive property, we can deduce that \(x \sim x\).
መልሱን አሳይ
Hint:
Let \(X = {\mathbb N} \cup \{ \sqrt{2}\, \}\) and define \(x \sim y\) if \(x + y \in {\mathbb N}\).
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Define a relation on \({\mathbb R}^2 \setminus \{ (0,0) \}\) by letting \((x_1, y_1) \sim (x_2, y_2)\) if there exists a nonzero real number \(\lambda\) such that \((x_1, y_1) = ( \lambda x_2, \lambda y_2)\). Prove that \(\sim\) defines an equivalence relation on \({\mathbb R}^2 \setminus (0,0)\). What are the corresponding equivalence classes? This equivalence relation defines the projective line, denoted by \({\mathbb P}({\mathbb R})\), which is very important in geometry.
Symbols used here
Ratios of sides in a right triangle; coordinates on the unit circle.
x belongs to A; every element of A is in B.
In either; in both; in A but not B.
2.71828…, the base whose exponential is its own derivative.
Naturals, integers, rationals, reals, complex numbers.
Marks the point where the statement has been established.
n divides a − b; a and b have the same remainder.
b is a multiple of a; the largest number dividing both.
A set with an operation; the do-nothing element; the element that undoes g.
Same structure; the group of cosets of a normal subgroup N.
The remainders 0…n−1 with clock arithmetic.
The set of morphisms; do g then f.
Questions people ask
What is a group, in plain words?
A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group — that is where the idea came from.
What is the difference between a ring and a field?
A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.
የራስዎን ይሞክሩ
Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.
በ Abstract Algebra
GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula