Introduction to Relations

Step 1: The range is formed from second components. Step 2: The second components are (2,4,6), so (3) is not in the range. Step 3: Avoid mixing up domain and range positions.

Step 1: The domain is the set of first components. Step 2: The first components in the given pairs are (a) and (b). Step 3: While finding domain, read only the first position.

Step 1: The range is made from second components. Step 2: The second components are (1,2,2), and repetition is not written in a set. Step 3: So the range is ({1,2}).

Step 1: In an inverse relation, the components of every pair are reversed. Step 2: ((1,2)) becomes ((2,1)), while diagonal pairs remain the same. Step 3: Reverse every pair one by one.

Step 1: In a symmetric relation, every pair is present with its reverse. Step 2: Therefore the inverse relation is the same as the original relation. Step 3: A symmetric relation can also be recognized by (R=R^{-1}).

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

Step 1: For reflexivity, ((1,1),(2,2),(3,3)) are required. Step 2: All three pairs are present in the relation. Step 3: Extra pairs do not remove the reflexive property.

Step 1: Symmetry requires the reverse of every pair. Step 2: ((1,2)) is paired with ((2,1)), and ((2,3)) with ((3,2)). Step 3: Identify symmetry by matching reverse pairs.

Step 1: Since ((1,2)) and ((2,3)) are present, transitivity requires ((1,3)). Step 2: ((1,3)) is missing in the relation. Step 3: In transitivity, when the middle element matches, check the required third pair.

Step 1: All pairs of (A\times A) are ((1,1),(1,2),(2,1),(2,2)). Step 2: The given relation contains all of them. Step 3: If all pairs are present, the relation is universal.

Step 1: A pair relating an element to itself has the form ((a,a)). Step 2: A relation containing only such pairs is the identity relation. Step 3: Remember identity relation as self-related pairs.

Step 1: The smallest reflexive relation contains only self-pairs for every element. Step 2: For four elements, there are four diagonal pairs. Step 3: In the smallest reflexive relation, the number of pairs equals the number of elements.

Step 1: The number of pairs in (A\times A) is (n(A)^2). Step 2: Here (n(A)=4), so (4^2=16). Step 3: For a set multiplied by itself, take the square of the number of elements.

Step 1: All given pairs are of the form ((a,a)). Step 2: Every element is related to itself and no extra pair is present. Step 3: Such a relation is called the identity relation.

Step 1: In symmetry, a pair appears with its reverse. Step 2: ((1,2)) and ((2,1)) are reverse pairs. Step 3: The presence of reverse pairs indicates symmetry.

Step 1: In a symmetric relation, every pair must have its reverse. Step 2: The reverse of ((3,5)) is ((5,3)). Step 3: For symmetry, interchange the first and second components.

Step 1: Transitivity requires ((a,c)) from ((a,b)) and ((b,c)). Step 2: From ((2,4)) and ((4,6)), the needed pair is ((2,6)). Step 3: Remove the common middle element and form the new pair.

Step 1: An empty relation has no ordered pair. Step 2: So it does not require pairs like ((a,a)). Step 3: In contrast, a reflexive relation requires all diagonal pairs.

Step 1: In an inverse relation, every pair is reversed. Step 2: ((1,2)) becomes ((2,1)), and ((2,1)) becomes ((1,2)). Step 3: So the inverse relation is the same relation.

Step 1: Symmetry requires ((2,1)) along with ((1,2)). Step 2: ((2,1)) is missing, so the relation is not symmetric. Step 3: Missing one reverse pair breaks symmetry.

Step 1: The domain is formed from first components. Step 2: The first components are (1,2,3). Step 3: Even in diagonal pairs, read the first component carefully.

Step 1: The range is formed from second components. Step 2: The second components are (1,2,3). Step 3: In an identity relation, domain and range can be the same.

Step 1: In (A\times B), the first component comes from (A) and the second from (B). Step 2: Hence the pairs are ((1,3)) and ((2,3)). Step 3: In Cartesian products, changing the order can change the answer.

Step 1: In (A\times B), the first component must come from (A). Step 2: In ((3,1)), the first component is (3), which is not in (A). Step 3: Position matters a lot in ordered pairs.

Step 1: Domain is formed from first components and range from second components. Step 2: The first components give (1,2), and the second components also give (1,2). Step 3: Even for a larger relation, find domain and range separately.