Introduction to Relations

Step 1: The number of pairs in (A\times B) is (n(A)n(B)). Step 2: Here (2\times3=6) pairs are formed. Step 3: For different sets multiply the numbers of elements.

Step 1: In (B\times A), the first component comes from (B) and the second from (A). Step 2: In ((3,1)), (3) is from (B) and (1) is from (A). Step 3: Always read the order carefully in Cartesian products.

Step 1: The domain is formed from the first components of ordered pairs. Step 2: The first components are (1,2,2), so the domain is ({1,2}). Step 3: Write repeated elements only once in a set.

Step 1: The range is formed from the second components of ordered pairs. Step 2: The second components are (3,3,4), so the range is ({3,4}). Step 3: Remember the different positions for domain and range.

Step 1: The domain contains first components. Step 2: The first components of the relation are (a,b,c), so (b) is in the domain. Step 3: Letters and numbers can both occur but position matters most.

Step 1: The range contains second components. Step 2: Here the second components are (2,3,2), so (2) belongs to the range. Step 3: To find the range look at the right-side components.

Step 1: In an inverse relation the two components of every pair are reversed. Step 2: ((1,4),(2,5),(3,6)) become ((4,1),(5,2),(6,3)). Step 3: While finding inverse do not miss any pair.

Step 1: To get (R), reverse every pair of (R^{-1}). Step 2: ((2,1)) gives ((1,2)) and ((3,2)) gives ((2,3)). Step 3: The inverse of the inverse is the original relation.

Step 1: (R=R^{-1}) means every pair is present with its reverse. Step 2: This is the main idea of a symmetric relation. Step 3: Symmetry can also be recognized through the inverse relation.

Step 1: Diagonal pairs reverse to themselves. Step 2: The non-diagonal pair ((1,2)) has its reverse ((2,1)) present. Step 3: In symmetry check the reverse of every non-diagonal pair.

Step 1: Every element is related to itself. Step 2: There is no pair with different components so it is the identity relation. Step 3: Identify the identity relation through diagonal pairs.

Step 1: (A\times A) contains all possible ordered pairs. Step 2: A relation containing all these pairs is called the universal relation. Step 3: When you see the full Cartesian product think of universal relation.

Step 1: The domain is formed from first components. Step 2: An empty relation has no pair so there is no first component. Step 3: The domain and range of an empty relation are both empty.

Step 1: The range is formed from second components. Step 2: An empty relation has no ordered pair so it has no second component. Step 3: In an empty relation both domain and range are empty.

Step 1: (A\times A) has (2^2=4) pairs. Step 2: A set with four elements has (2^4=16) subsets. Step 3: The total number of relations equals the number of subsets.

Step 1: (A\times A) has (4^2=16) pairs. Step 2: Each subset forms a relation so the total number of relations is (2^{16}). Step 3: For total relations on (n) elements use (2^{n^2}).

Step 1: (A\times B) has (2\times4=8) ordered pairs. Step 2: A relation is a subset of (A\times B), so total relations are (2^8). Step 3: For relations between two sets first count (mn) pairs.

Step 1: These four pairs are all possible pairs of (A\times A). Step 2: Since all are present, the relation is universal. Step 3: For a small set write the full Cartesian product and compare.

Step 1: An identity relation must have the diagonal pair for every element of the base set. Step 2: ((1,1)) and ((2,2)) are complete only for ({1,2}). Step 3: A property of a relation can change when the base set changes.

Step 1: All diagonal pairs are present so it is reflexive. Step 2: ((1,2)) and ((2,1)) are both present and transitivity also holds because the needed ((1,1)) and ((2,2)) are present. Step 3: For equivalence relation check all three properties separately.