Chapter 2 Relations Definition 2.1: An order pair (a, b) is a listing of the objects a and b in a prescribed order, with a appearing first and b appearing second. Two order pairs (a, b) and(c, d) are equal if only if a=c and b=d. {a,b}={b,a}, order pairs: (a, b*(b, a) unless a=b aa
Chapter 2 Relations Definition 2.1: An order pair (a,b) is a listing of the objects a and b in a prescribed order, with a appearing first and b appearing second. Two order pairs (a,b) and (c, d) are equal if only if a=c and b=d. {a,b}={b,a}， order pairs: (a,b)(b,a) unless a=b. (a,a)
Definition 2.2: The ordered n-tuple (alayy..,a) is the ordered collection that has a, as its first element, a, as its second element. and a as its nth element. Two ordered n-tuples are equal is only if each corresponding pair of their elements ia equal, i.e. a.a 1929 .,a)=(b,,b2s.,b,) if onlyif a =b. fo is Tor i=1,2,,n
Definition 2.2: The ordered n-tuple (a1 ,a2 ,…,an ) is the ordered collection that has a1 as its first element, a2 as its second element,…, and an as its nth element.Two ordered n-tuples are equal is only if each corresponding pair of their elements ia equal, i.e. (a1 ,a2 ,…,an )=(b1 ,b2 ,…,bn ) if only if ai=bi , for i=1,2,…,n
Definition 2. 3: Let a and b be two sets The Cartesian product of A and B, denoted by AXB, is the set of all ordered pairs(a2b) where a∈ A and b∈B. Hence A×B={(a,b)a∈ A and b∈B} Example: Let A=(1, 2, B=x, y, C=a, b, c) A×B={(1,x),(1,y),(2,x)2,y)}; B×A={(x,1),(x,2),(y,1),(y,2)}; B×A≠AXB commutative laws X
Definition 2.3: Let A and B be two sets. The Cartesian product of A and B, denoted by A×B, is the set of all ordered pairs ( a,b) where aA and bB. Hence A×B={(a, b)| aA and bB} Example: Let A={1,2}, B={x,y},C={a,b,c}. A×B={(1,x),(1,y),(2,x),(2,y)}; B×A={(x,1),(x,2),(y,1),(y,2)}; B×AA×B commutative laws ×
A×C={(1,a)(1,b)、(1,c),(2,a)2(2,b),2, c)} A×A={(1,1),(1,2),(2,1),(2,2)}。 A×=×A=8 Definition 2.4: LetA,A..a be n sets. The Cartesian product of A142…An denoted A1×A2×…XAn, is the set of all ordered n-tuples(a1,.,a)where a:∈A:fori=12n, Hence A1×A2×…)An={(a12a2…2n)a;∈A ,i=1,2,,n}
A×C={(1,a),(1,b),(1,c),(2,a),(2,b),(2, c)}; A×A={(1,1),(1,2),(2,1),(2,2)}。 A×=×A= Definition 2.4: Let A1 ,A2 ,…An be sets. The Cartesian product of A1 ,A2 ,…An , denoted by A1×A2×…×An , is the set of all ordered n-tuples (a1 ,a2 ,…,an ) where aiAi for i=1,2,…n. Hence A1×A2×…×An ={(a1 ,a2 ,…,an )|aiAi ,i=1,2,…,n}
Example: AXBXC=(,x, a), (1, x, b), (1, x, c),(1,y, a), (1’y,b),(1,y,c),(2,x,a)(2,x,b),(2,xc),(2y,a),(2,y,b) (2,y,c)} IfA= A for i=1,2,,n, then a1×A,×…× An by an Example: Let A represent the set of all students at an university, and let B represent the set of all course at the university. What is the cartesian product ofA×B? The Cartesian product of A XB consists of all the ordered pairs of the form(a, b), where a is a student at the university and b is a course offered at the university. The set A XB can be used to represent all possible enrollments of students in courses at the university
Example:A×B×C={(1,x,a),(1,x,b),(1,x,c),(1,y,a), (1,y,b), (1,y,c),(2,x,a),(2,x,b),(2,x,c),(2,y,a),(2,y,b), (2,y,c)}。 If Ai=A for i=1,2,…,n, then A1×A2×…×An by An . Example：Let A represent the set of all students at an university, and let B represent the set of all course at the university. What is the Cartesian product of A×B? The Cartesian product of A×B consists of all the ordered pairs of the form (a,b), where a is a student at the university and b is a course offered at the university. The set A×B can be used to represent all possible enrollments of students in courses at the university