A set is a collection where you can tell clearly whether something belongs to it, and a proposition is a statement that is clearly either true or false.
The intersection, the elements in both of two sets, and the union, the elements in at least one, become clear at a glance in a Venn diagram.
From a proposition of the form 'if p then q', swapping or negating the hypothesis and conclusion gives the converse, the inverse and the contrapositive, and of these only the contrapositive always has the same truth value as the original.
In the first picture, change the numbers of elements to check the formula for the size of a union; in the second, change whether p and q are true and compare the original statement with its contrapositive side by side.
Sets in a Venn diagram
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A Venn diagram showing the number of elements in each of its three regions and in the union as n(A), n(B) and n(A ∩ B) change
7
5
3
💡 Why subtract the intersection?
If 7 pupils are in the football club and 5 in the basketball club, that does not mean 12 pupils are in the two clubs. If 3 belong to both, those 3 have been counted twice. Taking the 3 away once gives the number in at least one club, which is 9. That is the formula for the number of elements in a union.
Number of elements in a union
Size of a union
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Subtract once the intersection that was counted twice
📐 Checking with the Venn diagram
①Elements only in A: n(A) − n(A ∩ B)
②Elements only in B: n(B) − n(A ∩ B)
③Elements in both: n(A ∩ B)
④Adding the three regions gives n(A) + n(B) − n(A ∩ B)
⑤Add the three numbers in the picture above and you always get the n(A ∪ B) on the bottom line
Complements and De Morgan's laws
De Morgan's law (1)
(A ∪ B)c = Ac ∩ Bc
The complement of a union is the intersection of the complements
De Morgan's law (2)
(A ∩ B)c = Ac ∪ Bc
The complement of an intersection is the union of the complements
💡 How to remember them
①Moving the complement inside the bracket swaps ∪ and ∩
②'Not in A or B' means the same as 'not in A and not in B'
③The difference of sets can be written the same way: A − B = A ∩ B^c
Propositions and contrapositives
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A table comparing the truth values of p → q and its contrapositive ¬q → ¬p as the truth of p and q changes
1
1
The proposition p → q
p → q is false only when p is true and q is false
If the hypothesis is true and the conclusion false, the proposition is false
🔍 Converse, inverse, contrapositive
①The converse q → p may or may not have the same truth value as the original
②The inverse ¬p → ¬q cannot be relied on to match the original either
③The contrapositive ¬q → ¬p always has the same truth value as the original
④If p → q is hard to prove directly, you can prove its contrapositive ¬q → ¬p instead
⑤If p → q is true, p is a sufficient condition for q and q is a necessary condition for p
Worked examples and unit test
Example 1
A and B are subsets of a universal set U with n(A) = 20, n(B) = 15 and n(A ∩ B) = 8. Find n(A ∪ B).
1
Use the formula for the number of elements in a union.
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
2
Substitute the values.
n(A ∪ B) = 20 + 15 − 8 = 27
▸ n(A ∪ B) = 27
There are 12 elements only in A, 7 only in B and 8 in both, and 12 + 7 + 8 = 27 confirms it.
Example 2
For a universal set U with n(U) = 30, n(A) = 18, n(B) = 14 and n(A ∩ B) = 7, find n(Ac ∩ Bc).
1
First find the number of elements in the union.
n(A ∪ B) = 18 + 14 − 7 = 25
2
By De Morgan's law, Ac ∩ Bc = (A ∪ B)c, so subtract from the universal set.
n(Ac ∩ Bc) = n(U) − n(A ∪ B) = 30 − 25 = 5
▸ n(Ac ∩ Bc) = 5
The question asks how many elements are in neither A nor B: the region outside both circles in a Venn diagram.
Summary
Grade-10 education-office assessment type
For real x, two conditions are p: 1 ≤ x ≤ 5 and q: x ≥ k. Find the largest real k for which p is sufficient for q.
①-1
②0
③1
④2
⑤3
▸ ③ 1
1
If p is sufficient for q, the truth sets satisfy P ⊆ Q.
P = {x | 1 ≤ x ≤ 5}, Q = {x | x ≥ k}, P ⊆ Q
2
For every element of P to satisfy x ≥ k, k must not exceed the minimum of P, which is 1.
k ≤ 1, so the maximum is k = 1
🎯 Points that often come up
①n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
②De Morgan's laws: moving the complement inside swaps ∪ and ∩
③p → q is false only when p is true and q is false
④A proposition and its contrapositive always have the same truth value
⑤When p → q is true, p is sufficient and q is necessary