Interactive Real Analysis - part of MathCS.org

• Real Analysis
• 1. Sets and Relations
• 1.1. Notation and Set Theory
• 1.2. Relations and Functions
• 1.3. Equivalence Relations and Classes
• 1.4. Natural Numbers, Integers, and Rational Numbers
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

Examples 1.2.3:

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Let A = {1, 2, 3, 4}, B = {14, 7, 234}, C = {a, b, c}, and R = real numbers. Define the following relations:

1. r relates A and B via: 1 ~ 234, 2 ~ 7, 3 ~ 14, 4 ~ 234, 2 ~ 234
2. f relates A and C via: {(1,c), (2,b), (3,a), (4,b)}
3. g relates A and C via: {(1,a), (2,a), (3,a)}
4. h relates R and itself via: {(x,sin(x))}

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1. The relation r is not a function, because the element 2 from the set A is associated with two elements from B.
2. The relation f is a function, because every element from A has exactly one relation from the set C.
3. The relation g is not a function, because the element {4} from the domain A has no element associated with it.
4. The relation h is a function with domain R, because every element {x} in R has exactly one element {sin(x)} associated with it.