Ordered Pair
If a pair of elements is listed in a specific order, then such a pair is called ordered pair.
An ordered pair is written by listing its two members in a specific order, separating them by a comma and enclosing the pair in parentheses.
The ordered pair of two elements a and b is denoted by (a,b): a being first element and b is second element.
Suppose A={red, blue} and B={bag, coat, shirt}, we can form 6 distinct ordered pairs of colored objects as (red,bag), (red,coat), (red,shirt), (blue,bag), (blue,coat), (blue,shirt).
Equality of Ordered Pairs
Two ordered pairs (a,b) and (c,d) are called equal, if their corresponding elements are equal i.e (a,b)=(c,d), iff a=c and b=d.
Ordered pairs (a,b) and (b,a) are different i.e. (a,b)≠(b,a), unless a=b
Ordered pairs may have the same first and second components such as (a,a), (5,5).
Also, {a,b}≠(a,b), because {a,b} is a set whereas (a,b) is an ordered pair.
Cartesian Product of Two Sets
Let A and B be any two non-empty sets, then the set of all ordered pairs (a,b) for all a∈A and b∈B is called the Cartesian product of A and B. It is written as A⨯B (read as A cross B).
Symbolically, A⨯B={(a,b): for all a∈A,b∈B}
Example: let A={1,2,3} and B={4,5}, then
A⨯B={(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)}
B⨯A={(4,1),(4,2),(4,3),(5,1),(5,2),(5,3)}
A⨯A={(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}
B⨯B={(4,4),(4,5),(5,4),(5,4)}
From the example we observe that,
(i) A⨯B ≠ B⨯A unless A=B
(ii) If A and B are finite sets, then n(A⨯B) = n(A)*n(B)
If there are p elements in A and q elements in B i.e. n(A)=p and n(B)=q, then n(A⨯B)=pq
(iii) If A and B are finite sets, then n(A⨯B) = n(B⨯A)
(iii) A⨯B = Ï• when one or both of A, B are empty sets.
(iv) A⨯B≠ Ï• iff A≠ Ï• and B≠ Ï•
(v) If A and B are non empty sets and either A or B is an infinite set, then A⨯B is also an infinite set.
Ordered Triplet
If A,B and C are three sets, then (a,b,c), where a∈A, b∈B and c∈C is called an ordered triplet.
Cartesian Product of Three Sets
If A, B and C are three sets, then Cartesian product of A,B and C is
A⨯B⨯C={(a,b,c): a∈A, b∈B, c∈C}
Properties of Cartesian Product of Sets
(i) A⨯B = B⨯A ⇔ A=B
(ii) A⨯B = A⨯C ⇔ B=C
(iii) A⨯(B∪C) = (A⨯B)∪(A⨯C)
(iv) A⨯(B∩C) = (A⨯B)∩(A⨯C)
(v) A⨯(B−C) = (A⨯B)−(A⨯C)
(vi) (A⨯B)∩(C⨯D) = (A∩C)⨯(B∩D)
(v) (A⨯B)∩(B⨯A) = (A∩B)⨯(B∩A)
(vi) If A⊂B and C⊂D, then A⨯C⊂B⨯D
If a pair of elements is listed in a specific order, then such a pair is called ordered pair.
An ordered pair is written by listing its two members in a specific order, separating them by a comma and enclosing the pair in parentheses.
The ordered pair of two elements a and b is denoted by (a,b): a being first element and b is second element.
Suppose A={red, blue} and B={bag, coat, shirt}, we can form 6 distinct ordered pairs of colored objects as (red,bag), (red,coat), (red,shirt), (blue,bag), (blue,coat), (blue,shirt).
Equality of Ordered Pairs
Two ordered pairs (a,b) and (c,d) are called equal, if their corresponding elements are equal i.e (a,b)=(c,d), iff a=c and b=d.
Ordered pairs (a,b) and (b,a) are different i.e. (a,b)≠(b,a), unless a=b
Ordered pairs may have the same first and second components such as (a,a), (5,5).
Also, {a,b}≠(a,b), because {a,b} is a set whereas (a,b) is an ordered pair.
Cartesian Product of Two Sets
Let A and B be any two non-empty sets, then the set of all ordered pairs (a,b) for all a∈A and b∈B is called the Cartesian product of A and B. It is written as A⨯B (read as A cross B).
Symbolically, A⨯B={(a,b): for all a∈A,b∈B}
Example: let A={1,2,3} and B={4,5}, then
A⨯B={(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)}
B⨯A={(4,1),(4,2),(4,3),(5,1),(5,2),(5,3)}
A⨯A={(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}
B⨯B={(4,4),(4,5),(5,4),(5,4)}
From the example we observe that,
(i) A⨯B ≠ B⨯A unless A=B
(ii) If A and B are finite sets, then n(A⨯B) = n(A)*n(B)
If there are p elements in A and q elements in B i.e. n(A)=p and n(B)=q, then n(A⨯B)=pq
(iii) If A and B are finite sets, then n(A⨯B) = n(B⨯A)
(iii) A⨯B = Ï• when one or both of A, B are empty sets.
(iv) A⨯B≠ Ï• iff A≠ Ï• and B≠ Ï•
(v) If A and B are non empty sets and either A or B is an infinite set, then A⨯B is also an infinite set.
Ordered Triplet
If A,B and C are three sets, then (a,b,c), where a∈A, b∈B and c∈C is called an ordered triplet.
Cartesian Product of Three Sets
If A, B and C are three sets, then Cartesian product of A,B and C is
A⨯B⨯C={(a,b,c): a∈A, b∈B, c∈C}
Properties of Cartesian Product of Sets
(i) A⨯B = B⨯A ⇔ A=B
(ii) A⨯B = A⨯C ⇔ B=C
(iii) A⨯(B∪C) = (A⨯B)∪(A⨯C)
(iv) A⨯(B∩C) = (A⨯B)∩(A⨯C)
(v) A⨯(B−C) = (A⨯B)−(A⨯C)
(vi) (A⨯B)∩(C⨯D) = (A∩C)⨯(B∩D)
(v) (A⨯B)∩(B⨯A) = (A∩B)⨯(B∩A)
(vi) If A⊂B and C⊂D, then A⨯C⊂B⨯D
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