Introduction to Relations

Step 1: For every number, (|a|=|a|), so the relation is reflexive. Step 2: Equality remains true when reversed, so it is symmetric. Step 3: If two equalities connect, the third equality also follows, so it is transitive.

Step 1: (a^2=a^2) is true for every (a). Step 2: If (a^2=b^2), then (b^2=a^2) is also true. Step 3: A chain of equalities gives (a^2=c^2), so all three properties hold.

Step 1: For non-zero (a), (a^2>0), so the relation is reflexive. Step 2: If (ab>0), then (ba>0), so it is symmetric. Step 3: The relation means same sign, and same sign passes through a chain.

Step 1: (a-a=0), which is not greater than (0), so the relation is not reflexive. Step 2: If (a>b) and (b>c), then (a>c), so it is transitive. Step 3: Greater-than type relations are generally not symmetric.

Step 1: If (a+b=0), then (b+a=0), so the relation is symmetric. Step 2: (a+a=0) is not true for every real (a), so it is not reflexive. Step 3: Symmetry alone is not enough for equivalence.

Step 1: For symmetry, the reverse ((2,1)) of ((1,2)) must be added. Step 2: ((3,3)) is its own reverse, so no extra pair is needed for it. Step 3: For the smallest relation add only compulsory pairs.

Step 1: Reflexivity needs ((1,1),(2,2),(3,3)). Step 2: ((1,1)) is already present in the relation. Step 3: Therefore only ((2,2)) and ((3,3)) must be added.

Step 1: Transitivity requires ((1,3)) from ((1,2)) and ((2,3)). Step 2: ((1,3)) is not in the relation. Step 3: Hence the minimum pair to add is ((1,3)).

Step 1: In ((1,2)) and ((2,1)), the middle element (2) matches. Step 2: Transitivity requires ((1,1)). Step 3: Reverse pairs often create a need for a diagonal pair.

Step 1: From ((1,2)) and ((2,3)), ((1,3)) is needed. Step 2: Then ((1,3)) and ((3,4)) require ((1,4)). Step 3: In a transitive extension, also check requirements created by newly added pairs.

Step 1: In the inverse relation, the components of every pair are reversed. Step 2: Therefore the range of the original relation becomes the domain of the inverse. Step 3: Remember that domain and range interchange in the inverse.

Step 1: An equivalence relation needs reflexive, symmetric, and transitive properties. Step 2: Here reflexivity is missing, so the condition is not satisfied. Step 3: Two properties are not enough; all three must be confirmed.

Step 1: An equivalence relation is transitive. Step 2: From ((1,2)) and ((2,3)), transitivity gives ((1,3)). Step 3: In equivalence relations use the given chain to find required pairs.

Step 1: An equivalence relation is symmetric. Step 2: If ((4,7)) is present, symmetry gives ((7,4)). Step 3: In equivalence relations always remember the reverse pair.

Step 1: Same parity means both numbers are odd or both are even. Step 2: (1) is odd, and the odd numbers in ({1,2,3,4}) are (1,3). Step 3: In an equivalence class, write the elements of the same group.

Step 1: On division by (3), the possible remainders are (0,1,2). Step 2: Elements with the same remainder go into the same class. Step 3: Hence (3) equivalence classes are formed.

Step 1: Dividing (2) by (3) gives remainder (2). Step 2: Dividing (5) by (3) also gives remainder (2). Step 3: Elements with the same remainder belong to the same equivalence class.

Step 1: Elements placed in the same block are related to each other. Step 2: (1) and (2) are in the same block ({1,2}). Step 3: Therefore ((1,2)) must be in the relation.

Step 1: In an equivalence relation formed by a partition, only elements in the same block are related. Step 2: (1) and (2) are in different blocks. Step 3: Therefore ((1,2)) will not be in the relation.

Step 1: Equivalence relations correspond to partitions of a set. Step 2: For two elements, the partitions are both separate or both together. Step 3: Hence there are (2) equivalence relations.