2.7 Ordered pairs
- 1. Introduction to set theory and to methodology and philosophy of mathematics and computer programming Ordered pairs An overview by Jan Plaza c 2017 Jan Plaza Use under the Creative Commons Attribution 4.0 International License Version of February 14, 2017
- 2. {x, y} – an unordered pair. {x, y} = {y, x} – not so with ordered pairs. A point on the plane is represented as an ordered pair of real numbers. 6 - r 5 3 5, 3 5, 3 = 3, 5
- 3. Definition (Kuratowski) The ordered pair with coordinates x, y , denoted x, y , is the set {{x}, {x, y}} {x, y} tells that x and y are the components of the ordered pair. {x} distinguishes the first component. Fact x, x = {{x}, {x, x}} = {{x}, {x}} = {{x}}
- 4. Proposition: a, b = c, d iff a = c and b = d.
- 5. Proposition: a, b = c, d iff a = c and b = d. Proof
- 6. Proposition: a, b = c, d iff a = c and b = d. Proof (←) (→)
- 7. Proposition: a, b = c, d iff a = c and b = d. Proof (←) Obvious. (→)
- 8. Proposition: a, b = c, d iff a = c and b = d. Proof (←) Obvious. (If the first object is known under the names a and c, and the second object is known under the names b and d, then a, b and c, d are ordered pairs formed from the same objects, and they are the same.) (→)
- 9. Proposition: a, b = c, d iff a = c and b = d. Proof (←) Obvious. (If the first object is known under the names a and c, and the second object is known under the names b and d, then a, b and c, d are ordered pairs formed from the same objects, and they are the same.) (→) on the next slide
- 10. Proposition: a, b = c, d iff a = c and b = d. Proof (→)
- 11. Proposition: a, b = c, d iff a = c and b = d. Proof (→) Assume that a, b = c, d .
- 12. Proposition: a, b = c, d iff a = c and b = d. Proof (→) Assume that a, b = c, d . Case: a = b. Case: a=b.
- 13. Proposition: a, b = c, d iff a = c and b = d. Proof (→) Assume that a, b = c, d . Case: a = b. a, b = c, d . Case: a=b.
- 14. Proposition: a, b = c, d iff a = c and b = d. Proof (→) Assume that a, b = c, d . Case: a = b. a, b = c, d . So, {{a}} = {{c}, {c, d}}. Case: a=b.
- 15. Proposition: a, b = c, d iff a = c and b = d. Proof (→) Assume that a, b = c, d . Case: a = b. a, b = c, d . So, {{a}} = {{c}, {c, d}}. So, {a} = {c} = {c, d}. Case: a=b.
- 16. Proposition: a, b = c, d iff a = c and b = d. Proof (→) Assume that a, b = c, d . Case: a = b. a, b = c, d . So, {{a}} = {{c}, {c, d}}. So, {a} = {c} = {c, d}. So, a = c = d. Case: a=b.
- 17. Proposition: a, b = c, d iff a = c and b = d. Proof (→) Assume that a, b = c, d . Case: a = b. a, b = c, d . So, {{a}} = {{c}, {c, d}}. So, {a} = {c} = {c, d}. So, a = c = d. Case: a=b. a, b = c, d .
- 18. Proposition: a, b = c, d iff a = c and b = d. Proof (→) Assume that a, b = c, d . Case: a = b. a, b = c, d . So, {{a}} = {{c}, {c, d}}. So, {a} = {c} = {c, d}. So, a = c = d. Case: a=b. a, b = c, d . So, {{a}, {a, b}} = {{c}, {c, d}}.
- 19. Proposition: a, b = c, d iff a = c and b = d. Proof (→) Assume that a, b = c, d . Case: a = b. a, b = c, d . So, {{a}} = {{c}, {c, d}}. So, {a} = {c} = {c, d}. So, a = c = d. Case: a=b. a, b = c, d . So, {{a}, {a, b}} = {{c}, {c, d}}. a=b.
- 20. Proposition: a, b = c, d iff a = c and b = d. Proof (→) Assume that a, b = c, d . Case: a = b. a, b = c, d . So, {{a}} = {{c}, {c, d}}. So, {a} = {c} = {c, d}. So, a = c = d. Case: a=b. a, b = c, d . So, {{a}, {a, b}} = {{c}, {c, d}}. a=b. So, {a, b}={c}.
- 21. Proposition: a, b = c, d iff a = c and b = d. Proof (→) Assume that a, b = c, d . Case: a = b. a, b = c, d . So, {{a}} = {{c}, {c, d}}. So, {a} = {c} = {c, d}. So, a = c = d. Case: a=b. a, b = c, d . So, {{a}, {a, b}} = {{c}, {c, d}}. a=b. So, {a, b}={c}. So, {a, b} = {c, d}.
- 22. Proposition: a, b = c, d iff a = c and b = d. Proof (→) Assume that a, b = c, d . Case: a = b. a, b = c, d . So, {{a}} = {{c}, {c, d}}. So, {a} = {c} = {c, d}. So, a = c = d. Case: a=b. a, b = c, d . So, {{a}, {a, b}} = {{c}, {c, d}}. a=b. So, {a, b}={c}. So, {a, b} = {c, d}. So, {c, d}={a}.
- 23. Proposition: a, b = c, d iff a = c and b = d. Proof (→) Assume that a, b = c, d . Case: a = b. a, b = c, d . So, {{a}} = {{c}, {c, d}}. So, {a} = {c} = {c, d}. So, a = c = d. Case: a=b. a, b = c, d . So, {{a}, {a, b}} = {{c}, {c, d}}. a=b. So, {a, b}={c}. So, {a, b} = {c, d}. So, {c, d}={a}. So, {a} = {c}.
- 24. Proposition: a, b = c, d iff a = c and b = d. Proof (→) Assume that a, b = c, d . Case: a = b. a, b = c, d . So, {{a}} = {{c}, {c, d}}. So, {a} = {c} = {c, d}. So, a = c = d. Case: a=b. a, b = c, d . So, {{a}, {a, b}} = {{c}, {c, d}}. a=b. So, {a, b}={c}. So, {a, b} = {c, d}. So, {c, d}={a}. So, {a} = {c}. So, a = c.
- 25. Proposition: a, b = c, d iff a = c and b = d. Proof (→) Assume that a, b = c, d . Case: a = b. a, b = c, d . So, {{a}} = {{c}, {c, d}}. So, {a} = {c} = {c, d}. So, a = c = d. Case: a=b. a, b = c, d . So, {{a}, {a, b}} = {{c}, {c, d}}. a=b. So, {a, b}={c}. So, {a, b} = {c, d}. So, {c, d}={a}. So, {a} = {c}. So, a = c. So, b = d.
- 26. Fact x, y = y, x iff x = y.
- 27. Thinking about program variables and their values. (In two different ways.) 1. A variable has a value: VARIABLE-NAME, VALUE . 2. A variable refers to a memory address. The memory address stores a value. VARIABLE-NAME, MEMORY-ADDRESS together with MEMORY-ADDRESS, VALUE .