MH Board · Chapter

Sets and Relations

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subsets, counting

\varnothing

and itself

Locked

Union and intersection

A \cup B

: in either;

A \cap B

: in bothLocked

Difference and complement

A - B

: in

A

but not

B

;

A'

: in

U

but not

A

Locked

De Morgan's laws

(A \cup B)' = A' \cap B'

and

(A \cap B)' = A' \cup B'

Locked

Counting with two sets $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ ; "only A" and "neither" from the Venn diagramLocked
Counting with three setsAdd the singles, subtract the pairs, add back the triple; fill the Venn diagram from the centre outLocked
Least and greatest overlap $n(A \cap B)$ is at least $n(A) + n(B) - n(U)$

Interval notationSquare bracket: the end is in; round bracket: it isn't; $\infty$ always gets a round oneLocked
Operations on intervalsUnion, intersection, difference and complement of intervals, read off the number lineLocked
Solving inequalities as intervalsLinear: flip the sign when dividing by a negative; quadratic and rational: test the sign between critical pointsLocked

Ordered pairs and the Cartesian product $(a, b) = (c, d)$ only when $a = c$ and $b = d$ ; $n(A \times B) = n(A)\,n(B)$ Locked
Relations: domain, co-domain and rangeA relation is a subset of $A \times B$ ; its domain is the first entries used, its range the secondLocked
How many relationsEvery subset of $A \times B$ is a relation: $2^{mn}$ of themLocked
Reflexive, symmetric, transitiveCheck every $(a, a)$ ; every $(a, b)$ has its $(b, a)$ ; every chain $(a, b), (b, c)$
Equivalence relationsReflexive, symmetric and transitive together; it splits the set into classesLocked

and at most the smaller of

n(A)

,

n(B)

Locked

closes with

(a, c)

Locked