Exclusive Disjunction

Transcription

1 Exclusive Disjunction Recall A statement is a declarative sentence that is either true or false, but not both. If we have a declarative sentence s, p: s is true, and q: s is false, can we rewrite s is a statement into symbols?

2 Exclusive Disjunction Recall A statement is a declarative sentence that is either true or false, but not both. If we have a declarative sentence s, p: s is true, and q: s is false, can we rewrite s is a statement into symbols? (p q) ( (p q)) is one direct answer.

3 Exclusive Disjunction Recall A statement is a declarative sentence that is either true or false, but not both. If we have a declarative sentence s, p: s is true, and q: s is false, can we rewrite s is a statement into symbols? (p q) ( (p q)) is one direct answer. The textbook uses p q to simplify the notation, and calls it the exclusive disjunction of p and q.

4 Exclusive Disjunction Recall A statement is a declarative sentence that is either true or false, but not both. If we have a declarative sentence s, p: s is true, and q: s is false, can we rewrite s is a statement into symbols? (p q) ( (p q)) is one direct answer. The textbook uses p q to simplify the notation, and calls it the exclusive disjunction of p and q. What s the truth value?

5 Exclusive Disjunction (Continued) Now we want to find the truth table for (p q) ( (p q)), the exclusive disjunction of p and q. Beginning with p q p q (p q) (p q) ( (p q)) T T T F F T F F, we fill the third column to have

6 Exclusive Disjunction (Continued) Now we want to find the truth table for (p q) ( (p q)), the exclusive disjunction of p and q. Beginning with p q p q (p q) (p q) ( (p q)) T T T F F T F F, we fill the third column to have p q p q (p q) (p q) ( (p q)) T T T T F T F T T F F F.

7 Exclusive Disjunction (Continued) Then we fill the fourth column, getting p q p q (p q) (p q) ( (p q)) T T T F T F T T F T T T F F F T.

8 Exclusive Disjunction (Continued) Then we fill the fourth column, getting p q p q (p q) (p q) ( (p q)) T T T F T F T T F T T T F F F T. Finally there is p q p q (p q) (p q) ( (p q)) T T T F F T F T T T F T T T T F F F T F.

9 Exclusive Disjunction (Continued) Then we fill the fourth column, getting p q p q (p q) (p q) ( (p q)) T T T F T F T T F T T T F F F T. Finally there is p q p q (p q) (p q) ( (p q)) T T T F F T F T T T F T T T T F F F T F. Conclusion: p q is true iff p and q have opposite truth values.

10 Back to the Initial Problem Fill the truth table p q p q p q p q T T T F F T F F, you will have

11 Back to the Initial Problem Fill the truth table p q p q p q p q T T T F F T F F, you will have p q p q p q p q T T T T F F T F F T F T F T F T T F F F F F T T. Each row has two T s, explaining the answer.

12 Summary You are supposed to know how to create truth tables for compound statements, and determine their truth values, based on the truth values of their perspective generators (p, q, etc.).

13 1.1: Introduction to Sets Jiakun Pan Jan 18, 2019

14 Sets A set is a collection of items. We use curly brackets {, } to contain it. Each of these items are called an element or member of the set.

15 Sets A set is a collection of items. We use curly brackets {, } to contain it. Each of these items are called an element or member of the set. For example, the set {1, 2, 3} is the collection of numbers 1, 2, and 3. These numbers are the elements or members of the set.

16 Sets A set is a collection of items. We use curly brackets {, } to contain it. Each of these items are called an element or member of the set. For example, the set {1, 2, 3} is the collection of numbers 1, 2, and 3. These numbers are the elements or members of the set. Order doesn t matter, so {1, 2, 3} = {3, 2, 1}.

17 Sets A set is a collection of items. We use curly brackets {, } to contain it. Each of these items are called an element or member of the set. For example, the set {1, 2, 3} is the collection of numbers 1, 2, and 3. These numbers are the elements or members of the set. Order doesn t matter, so {1, 2, 3} = {3, 2, 1}. Every element of a set must be unique. In other words, things like {1, 2, 2} are not considered to be sets.

18 Sets A set is a collection of items. We use curly brackets {, } to contain it. Each of these items are called an element or member of the set. For example, the set {1, 2, 3} is the collection of numbers 1, 2, and 3. These numbers are the elements or members of the set. Order doesn t matter, so {1, 2, 3} = {3, 2, 1}. Every element of a set must be unique. In other words, things like {1, 2, 2} are not considered to be sets.

19 Representing Sets There are two ways to write out a set. Roster notation lists all elements of the set, like {1, 2, 3}.

20 Representing Sets There are two ways to write out a set. Roster notation lists all elements of the set, like {1, 2, 3}. Set-builder notation describes all elements instead, so you may also write {x x is an integer greater than 1 2 and less than π}, or {Integer x 1 2 < x < π}, for the same set.

21 Representing Sets There are two ways to write out a set. Roster notation lists all elements of the set, like {1, 2, 3}. Set-builder notation describes all elements instead, so you may also write {x x is an integer greater than 1 2 and less than π}, or {Integer x 1 2 < x < π}, for the same set. Conventionally we use uppercase letters A, B,... to denote sets, and use lowercase letters a, b,... to denote elements. If a set A contains an element a (or a is in A), we write a A.

22 Subsets Given sets A and B, we call A is a subset of B (or B contains A), if statement x B is true for all x A. Write it A B.

23 Subsets Given sets A and B, we call A is a subset of B (or B contains A), if statement x B is true for all x A. Write it A B. If A B, and B A, then A = B.

24 Subsets Given sets A and B, we call A is a subset of B (or B contains A), if statement x B is true for all x A. Write it A B. If A B, and B A, then A = B. Furthermore, if A B, but A B, then we also call A is a proper subset of B, and write A B.

25 Subsets Given sets A and B, we call A is a subset of B (or B contains A), if statement x B is true for all x A. Write it A B. If A B, and B A, then A = B. Furthermore, if A B, but A B, then we also call A is a proper subset of B, and write A B. We negate all symbols with a slash /, so a / A means a is not in A, A B says A is not a subset of B, and A B stands for A is not a proper subset of B.

26 Subsets Given sets A and B, we call A is a subset of B (or B contains A), if statement x B is true for all x A. Write it A B. If A B, and B A, then A = B. Furthermore, if A B, but A B, then we also call A is a proper subset of B, and write A B. We negate all symbols with a slash /, so a / A means a is not in A, A B says A is not a subset of B, and A B stands for A is not a proper subset of B. Transitivity Law: A B C implies A C. Same for.

27 The Empty Set If a set has not elements, then we call the set the empty set, and write it as {} or.

28 The Empty Set If a set has not elements, then we call the set the empty set, and write it as {} or. By definition, the empty set is a subset of all sets.

29 The Empty Set If a set has not elements, then we call the set the empty set, and write it as {} or. By definition, the empty set is a subset of all sets. Example Find all subsets of {T, A, M, U}.

30 The Empty Set If a set has not elements, then we call the set the empty set, and write it as {} or. By definition, the empty set is a subset of all sets. Example Find all subsets of {T, A, M, U}. There are 16 subsets in total, which are {T, A, M, U}, {A, M, U}, {T, M, U}, {T, A, U}, {T, A, M}, {M, U}, {T, M}, {A, M}, {T, A}, {T, U}, {A, U}, {T }, {A}, {M}, {U}, and.

31 The Universal Set and Venn Diagram The universal set is opposite to the empty set. Its elements depend on topic that we are discussing. All sets are subsets of it. As a convention, we use letter U to represent the universal set.

32 The Universal Set and Venn Diagram The universal set is opposite to the empty set. Its elements depend on topic that we are discussing. All sets are subsets of it. As a convention, we use letter U to represent the universal set. A Venn diagram is a way of visualizing (NOT writing!) sets. The universal set is represented by a rectangle, while others by circles. A B U Above shows the relationship between A and B within a fixed universal set U, which is B A, and more accurately, B A.

33 Set Operations Just like we use connectives to generate new statements, we can also generate new sets with given sets, and the tools here are set operations.

34 Set Operations Just like we use connectives to generate new statements, we can also generate new sets with given sets, and the tools here are set operations. There are three types of operations, too. Intersection - Union - Complement - c

35 Intersection & Union Given two sets A and B, we write A B for the intersection of A and B, which is the set that contains elements in both of the two sets, and nothing else.

36 Intersection & Union Given two sets A and B, we write A B for the intersection of A and B, which is the set that contains elements in both of the two sets, and nothing else. Accordingly, we denote the union of A and B by A B, the set that consists of elements belonging to either A or B.

37 Intersection & Union Given two sets A and B, we write A B for the intersection of A and B, which is the set that contains elements in both of the two sets, and nothing else. Accordingly, we denote the union of A and B by A B, the set that consists of elements belonging to either A or B. By definition, you can see A B = B A; A B = B A; A B A A B; and A B B B A.

38 Example Example Let U be {(x, y) x, y are real numbers}. Write A = {(x, y) U x = 0}, and B = {(x, y) U y = 1}. Question: what are A B and A B?

39 Example Example Let U be {(x, y) x, y are real numbers}. Write A = {(x, y) U x = 0}, and B = {(x, y) U y = 1}. Question: what are A B and A B? Answer: A B = {(x, y) U x = 0 and y = 1} = {(0, 1)}; A B = {(x, y) U x = 0 or y = 1}.