Relations and Functions

If set A contains m elements and set B contains n elements, then the set A x B will have

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m+n

m.n

m-n

m.m

n.n

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Any subset of ordered pairs in A x B is called a relation from

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A to B

B to A

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A x B

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Let A = {1, 2, 3} and B = {2, 4}, then A x B =

{(2,1), (4,1), (2,2), (4,2), (2,3), (4,3)}

{(2,1), (4,1), (2,2), (4,2), (2,3)}

{(1,2), (1,4), (2,2), (2,4), (3,2), (3,4)}

{(1,2), (1,4), (2,2), (2,4), (3,2)}

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Function from A to B

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A collection of ordered pairs (from the set A x B) constitute a special relation from A to B, which is called a function from A to B if we select the ordered pairs in such a way that:

their first elements constitute the entire set A

no two distinct pairs have the same first element

their second elements constitute the entire set B

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Relations from A to B

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Let A = {1, 2, 3, 4} and B = {2, 4, 5}

The following are all possible relations from A to B

R₁ = {(1, 2), (1, 5), (2, 2), (3, 4), (3, 5), (4, 5)}
R₂ = {(1, 4), (4, 2), (4, 5)}
R₃ = {(3, 2), (3, 4), (3, 5), (1, 4)}
R₄ = {(1, 4), (2, 5), (3, 2), (4, 4)}
R₅ = {(1, 2), (2, 5), (3, 4), (4, 4)}

True

False

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Product with empty set

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Let A = {1, 2, 3}, B = ∅, Then A x B =

{(1,1), (2,2), (3,3)}

∅

{1, 2, 3}

{(1,∅),(2,∅),(3,∅)}

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Domain of a Relation

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In any relation (in the form of a set of ordered pairs), the set consisting of the ________ element of each pair constitutes the domain of the relation.

first

second

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Given a domain of R, tell if it is a function

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Let A = {1, 2, 3, 4} and B = {2, 4, 5}

The following are all possible relations from A to B

R₁ = {(1, 2), (1, 5), (2, 2), (3, 4), (3, 5), (4, 5)}
R₂ = {(1, 4), (4, 2), (4, 5)}
R₃ = {(3, 2), (3, 4), (3, 5), (1, 4)}
R₄ = {(1, 4), (2, 5), (3, 2), (4, 4)}
R₅ = {(1, 2), (2, 5), (3, 4), (4, 4)}

Which of these relations represent a function

R₁ (1) a function
R₂ (2) a function
R₃ (3) a function
R₄ (4) a function
R₅ (5) a function

Please drag and drop the selected option in the right place or type it instead

represents

does not represent

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Number of Elements in A × B

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If set \(A\) contains \(m\) elements and set \(B\) contains \(n\) elements, how many ordered pairs are in the Cartesian product \(A \times B\)?

\(m + n\)

\(m \times n\)

\(m^n\)

\(n^m\)

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Definition of a Relation

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What is a relation from set \(A\) to set \(B\)?

Any subset of the Cartesian product \(A \times B\)

The entire set \(A \times B\)

Any subset of \(A \cup B\)

Any mapping from \(A\) onto \(B\)

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Computing A × B with Specific Sets

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Let \(A = \{1, 2, 3\}\) and \(B = \{2, 4\}\). Which of the following is \(A \times B\)?

\(\{(1,2), (1,4), (2,2), (2,4), (3,2), (3,4)\}\)

\(\{(1,2), (2,4), (3,2)\}\)

\(\{(2,1), (4,1), (2,2), (4,2), (2,3), (4,3)\}\)

\(\{1,2,3,2,4\}\)

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Condition for a Relation to Be a Function

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What condition must a relation from \(A\) to \(B\) satisfy to be called a function?

Every element of \(A\) must appear as the first element in exactly one ordered pair

Every element of \(B\) must appear as the second element in at least one ordered pair

No two ordered pairs may share the same second element

The relation must be equal to \(A \times B\)

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Identifying a Function from a Set of Ordered Pairs

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Let \(A = \{1, 2, 3, 4\}\) and \(B = \{2, 4, 5\}\). Which of the following relations from \(A\) to \(B\) is a function?

\(R_1 = \{(1, 2), (1, 5), (2, 2), (3, 4), (3, 5), (4, 5)\}\)

\(R_4 = \{(1, 4), (2, 5), (3, 2), (4, 4)\}\)

\(R_2 = \{(1, 4), (4, 2), (4, 5)\}\)

\(R_3 = \{(3, 2), (3, 4), (3, 5), (1, 4)\}\)

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Cartesian Product with the Empty Set

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Let \(A = \{1, 2, 3\}\) and \(B = \varnothing\). What is \(A \times B\)?

\(\{(1, \varnothing), (2, \varnothing), (3, \varnothing)\}\)

\(\varnothing\)

\(\{\varnothing\}\)

\(\{1, 2, 3\}\)

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Domain of a Relation

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In a relation defined as a set of ordered pairs, what constitutes the domain of the relation?

The set of all second elements from each pair

The set of all first elements from each pair

The union of all first and second elements

The entire Cartesian product from which the relation is taken

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Range of a Relation

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For the relation \(R = \{(1, 2), (3, 4), (5, 2), (7, 8)\}\), what is the range?

\(\{1, 3, 5, 7\}\)

\(\{2, 4, 8\}\)

\(\{2, 4, 2, 8\}\)

\(\{1, 2, 3, 4, 5, 7, 8\}\)

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Function Restriction: Uniqueness of Images

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Why is \(R = \{(1, 2), (1, 3), (2, 5)\}\) not a function from \(\{1, 2\}\) to \(\{2, 3, 5\}\)?

It is not a subset of \(A \times B\)

The element 1 in the domain has two different images (2 and 3)

The element 5 in the codomain has no pre-image

The domain element 2 appears only once

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Determining if a Relation is a Function

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Let \(A = \{1, 2, 3, 4\}\). Which of the following relations on \(A\) (i.e., from \(A\) to \(A\)) is a function?

\(\{(1, 2), (2, 3), (3, 4), (4, 1)\}\)

\(\{(1, 1), (1, 2), (2, 3), (3, 4)\}\)

\(\{(1, 2), (2, 2), (3, 2), (4, 2), (2, 3)\}\)

\(\{(1, 2), (2, 3), (3, 4)\}\)

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Size of the Power Set of A × B

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If |A|=m and |B|=n, how many distinct relations from A to B are possible?

\(2^{m+n}\)

\(2^{mn}\)

\(m^n\)

\(n^m\)

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Empty Relation as a Function

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Let \(A = \{1, 2\}\) and \(B = \{3, 4\}\). Is the empty relation \(R = \varnothing\) a function from \(A\) to \(B\)?

Yes, because the empty set is a subset of \(A \times B\)

No, because it violates the condition that every element of \(A\) must have an image

Yes, but only if \(A\) is also empty

No, because a function must contain at least one ordered pair

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If set A contains m elements and set B contains n elements, then the set A x B will have

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Explanation

Any subset of ordered pairs in A x B is called a relation from

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Explanation

A x B

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Explanation

Function from A to B

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Explanation

Relations from A to B

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Explanation

Product with empty set

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Explanation

Domain of a Relation

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Given a domain of R, tell if it is a function

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Number of Elements in A × B

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Definition of a Relation

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Explanation

Computing A × B with Specific Sets

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Explanation

Condition for a Relation to Be a Function

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Explanation

Identifying a Function from a Set of Ordered Pairs

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Explanation

Cartesian Product with the Empty Set

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Explanation

Domain of a Relation

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Explanation

Range of a Relation

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Explanation

Function Restriction: Uniqueness of Images

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Explanation

Determining if a Relation is a Function

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Explanation

Size of the Power Set of A × B

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Explanation

Empty Relation as a Function

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Explanation

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