Binary Relations

Binary Relations: The Secret Language of Connections

The Story of Matchmaking

Imagine you’re at a big party where everyone wears a name tag. Now, some people know each other, and some don’t. A binary relation is like having a magical notebook that writes down every time two people are connected somehow.

Think of it like this: You have two groups of kids. Group A has kids named Alice, Bob, and Charlie. Group B has toys: Ball, Doll, and Car. When a kid picks a favorite toy, we draw an arrow from the kid to the toy. That collection of arrows? That’s a binary relation!

What is a Binary Relation?

A binary relation connects elements from one set to elements of another set (or the same set).

Simple Example:

• Set A = {1, 2, 3} (kids’ ages)
• Set B = {Red, Blue, Green} (favorite colors)

If age 1 likes Red, age 2 likes Blue, and age 3 likes Red too, our relation R is:

R = {(1, Red), (2, Blue), (3, Red)}

Each pair (a, b) means: “a is related to b”

Real Life Examples:

• “is parent of” connects parents to children
• “is taller than” connects people by height
• “likes” connects you to your favorite foods

graph TD
    A["1"] -->|likes| B["Red"]
    C["2"] -->|likes| D["Blue"]
    E["3"] -->|likes| B

Domain of a Relation

The domain is like asking: “Who sent a message?”

It’s the set of all first elements in our pairs.

Example:

If R = {(1, Red), (2, Blue), (3, Red)}

Domain = {1, 2, 3}

These are all the elements that start a connection.

Think of it this way:

• You’re throwing paper airplanes
• The domain is all the kids who threw airplanes
• It doesn’t matter where they landed
• Just who threw them!

Range of a Relation

The range is like asking: “Who received a message?”

It’s the set of all second elements that actually got picked.

Example:

If R = {(1, Red), (2, Blue), (3, Red)}

Range = {Red, Blue}

Notice: Green is NOT in the range because nobody picked it!

Think of it this way:

• Paper airplanes landing
• Range = only the spots where airplanes actually landed
• Not every possible spot, just the ones hit!

graph TD
    subgraph Domain
    A["1"]
    B["2"]
    C["3"]
    end
    subgraph Range
    D["Red"]
    E["Blue"]
    end
    A --> D
    B --> E
    C --> D

Codomain

The codomain is the entire “landing zone” — all possible destinations, whether airplanes land there or not.

Example:

• Set A = {1, 2, 3}
• Set B = {Red, Blue, Green} ← This is the CODOMAIN
• R = {(1, Red), (2, Blue), (3, Red)}

Codomain = {Red, Blue, Green}

Range = {Red, Blue} (only what was actually used)

The Big Difference:

Real Life:

• Menu at restaurant = Codomain (all possible dishes)
• What you actually ordered = Range

Inverse Relation

The inverse relation is like pressing “reverse” on everything!

If R goes from A to B, then R⁻¹ goes from B to A.

Example:

If R = {(1, Red), (2, Blue), (3, Red)}

Then R⁻¹ = {(Red, 1), (Blue, 2), (Red, 3)}

We just flip every pair!

Think of it this way:

• Normal: “Kids who like colors”
• Inverse: “Colors liked by which kids”

graph TD
    subgraph Original R
    A1["1"] --> B1["Red"]
    end
    subgraph Inverse R⁻¹
    B2["Red"] --> A2["1"]
    end

Real Life:

• “is parent of” → inverse is “is child of”
• “sent message to” → inverse is “received message from”

Composition of Relations

Composition is like a chain reaction or relay race!

If R goes from A to B, and S goes from B to C, then S ∘ R (read “S composed with R”) goes directly from A to C.

Example:

• R = {(1, a), (2, b)} — from numbers to letters
• S = {(a, X), (b, Y)} — from letters to symbols

S ∘ R = {(1, X), (2, Y)}

How? Follow the chain:

• 1 → a → X (so 1 connects to X)
• 2 → b → Y (so 2 connects to Y)

Think of it like:

• You tell a secret to Friend A
• Friend A tells it to Friend B
• The composition is: You → Friend B (skipping the middle!)

graph TD
    A["1"] -->|R| B["a"]
    B -->|S| C["X"]
    D["2"] -->|R| E["b"]
    E -->|S| F["Y"]

Step by Step:

1. Start with an element from the first set
2. Follow R to get to the middle set
3. Then follow S to get to the final set
4. Write down (start, finish)

Summary: All Concepts Together

Let’s use one complete example:

Set A = {Tom, Jerry} Set B = {Pizza, Burger, Salad} Set C = {Yummy, Okay, Gross}

Relation R (who eats what) = {(Tom, Pizza), (Tom, Burger), (Jerry, Salad)}

Relation S (food ratings) = {(Pizza, Yummy), (Burger, Okay), (Salad, Gross)}

Quick Memory Tricks

• Domain = Departure point (where arrows start)
• Range = Received (where arrows actually land)
• Codomain = Complete possibilities (all landing spots)
• Inverse = Invert the arrows (flip the pairs)
• Composition = Chain together (follow the path)

You Did It!

Now you understand binary relations like a pro! Remember:

1. A binary relation is just pairs showing connections
2. Domain = all the starting points
3. Range = all the actual endpoints
4. Codomain = all possible endpoints
5. Inverse = flip every connection
6. Composition = chain two relations together

You’re now ready to see these connections everywhere — from friend networks to family trees to your favorite foods!