Concept of Ordered Pairs
Before understanding the concept of Relations, it’s important to grasp the concept of ordered pairs and mathematical open sentences. This understanding will facilitate comprehension of Relations.
Ordered Pairs
Definition: A pair of two objects ( a ) and ( b ), denoted by ( (a,b) ), is called an ordered pair if ( (a,b) ) observes the order.
From the definition of ordered pairs, it can be understood that ( (a, b) \neq (b, a) ), where ( a \neq b ), because the first component of ( (a,b) ) is occupied by object ( a ), while the first component of ( (b,a) ) is occupied by ( b ).
Mathematical Open Sentences
Mathematical sentences with variables ( x ) and ( y ) are usually expressed as ( P(x,y) ). Here, we define Mathematical Open Sentences.
- Definition: A mathematical sentence ( P(x,y) ) is called a Mathematical Open Sentence with variables ( x ) and ( y ) if the truth value of ( P(x,y) ) cannot be determined.
The truth value of a mathematical open sentence ( P(x,y) ) depends greatly on its variables.
Example
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( P(x,y) ): “x is less than y”. In the open sentence “x is less than y”, it will be true if we substitute ( x = 2 ) and ( y = 3 ), or ( P(2,3) ) is true. But it will be false if we substitute ( x = 4 ) and ( y = 1 ), or ( P(4,1) ) is false.
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( P(a,b) ): “a + b = 8”. In the open sentence “a + b = 8”, it will be true if we substitute ( a = 5 ) and ( b = 3 ), or ( P(5,3) ) is true. But it will be false if we substitute ( a = 2 ) and ( b = 3 ), or ( P(2,3) ) is false.