Symmetric relations
R is symmetric if whenever (a, b) ∈ R, the reverse pair (b, a) is also in R. Every pair needs its mirror pair.
What symmetric means
A relation R on a set A is symmetric if (a, b) ∈ R implies (b, a) ∈ R for all a, b in A. In words: if a is related to b, then b must be related to a. 'Is a cousin of' is symmetric: if Kavin is Divya's cousin, Divya is Kavin's cousin. 'Is the father of' is not symmetric. On a finite set, check every pair (a, b) in R and look for its reverse (b, a). On an infinite set, start from a general pair (a, b) ∈ R and prove that (b, a) ∈ R.
Common relations: symmetric or not?
| Relation | On the set | Symmetric? | Reason |
|---|---|---|---|
| a = b | R | Yes | a = b gives b = a |
| a ≤ b | R | No | 2 ≤ 3, but 3 ≤ 2 is false |
| L₁ ⊥ L₂ | Lines in a plane | Yes | If L₁ ⊥ L₂, then L₂ ⊥ L₁ |
| a divides b | N | No | 2 divides 6, but 6 does not divide 2 |
| a − b is divisible by 5 | Z | Yes | b − a = −(a − b) is also divisible by 5 |
| x is the brother of y | People | No | x can be the brother of his sister y, but y is not his brother |
Example 1: a small set
Let A = {1, 2, 3} and R = {(1, 2), (2, 1), (1, 3), (3, 3)}. Is R symmetric? If not, make it symmetric by adding the fewest pairs.
- Rule: R is symmetric if (a, b) ∈ R ⇒ (b, a) ∈ R.
- (1, 2) ∈ R and its reverse (2, 1) ∈ R.
- (2, 1) ∈ R and its reverse (1, 2) ∈ R.
- (3, 3) is its own reverse, so it causes no problem.
- (1, 3) ∈ R, but its reverse (3, 1) ∉ R.
- So R is not symmetric. Adding (3, 1) fixes it.
R is not symmetric, because (1, 3) ∈ R but (3, 1) ∉ R. The smallest symmetric relation containing R is R ∪ {(3, 1)}.
Example 2: a proof on integers (board style)
Show that the relation R = {(a, b): a − b is divisible by 4} on the set Z of integers is symmetric.
- Rule: show that (a, b) ∈ R ⇒ (b, a) ∈ R.
- Let (a, b) ∈ R. Then a − b is divisible by 4.
- So a − b = 4k for some integer k.
- Then b − a = −(a − b) = −4k = 4(−k).
- −k is also an integer, so b − a is divisible by 4.
- So (b, a) ∈ R.
R is symmetric.
Example 3: a counterexample on real numbers
Show that R = {(a, b): a < b²} on the set R of real numbers is not symmetric.
- Rule: one pair (a, b) ∈ R with (b, a) ∉ R is enough.
- Take a = 1, b = 2: is 1 < 2² = 4? Yes, so (1, 2) ∈ R.
- Reverse: is 2 < 1² = 1? No.
- So (2, 1) ∉ R.
R is not symmetric, since (1, 2) ∈ R but (2, 1) ∉ R.
In a cricket tournament, 'played a match against' is symmetric: if Chennai played Mumbai, Mumbai played Chennai. 'Follows on social media' is not symmetric, because you can follow someone who does not follow you back.
1 marks: “Let A = {1, 2, 3} and R = {(1, 2), (2, 1), (2, 3)}. Which pair must be added to make R symmetric?” (3, 2), the reverse of (2, 3). The pairs (1, 2) and (2, 1) already match.
2 marks: “Show that the relation R on Z given by R = {(a, b): a + b is even} is symmetric.” Let (a, b) ∈ R, so a + b is even. Since b + a = a + b, b + a is also even. So (b, a) ∈ R, and R is symmetric.
1 marks: “Is the relation 'a divides b' on N symmetric?” No. 2 divides 4, but 4 does not divide 2. So (2, 4) ∈ R but (4, 2) ∉ R.
Symmetric = two-way street. For every arrow from a to b, look for the arrow from b to a.