Practice Disjunction in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A disjunction P∨QP \vee Q is a compound statement that is true whenever at least one of its parts is true.

At least one must be true. Logical OR is inclusive — "P or Q or both" — unlike the exclusive everyday "either/or."

Showing a random 20 of 50 problems.

Example 1

easy
Evaluate: '3>53>5 OR 2<42<4'.

Example 2

medium
P∨F≡P \vee F \equiv ____ (identity law).

Example 3

challenge
How many of the 23=82^3=8 rows make P∨Q∨RP\vee Q\vee R true, and why?

Example 4

hard
Prove (P⇒Q)≡(¬P∨Q)(P \Rightarrow Q) \equiv (\neg P \vee Q) via truth table.

Example 5

medium
Solve 'x2−5x+6=0x^2 - 5x + 6 = 0' and explain why the answer is naturally an OR.

Example 6

medium
Use a truth table to verify De Morgan's Law: ¬(p∨q)≡¬p∧¬q\neg(p \lor q) \equiv \neg p \land \neg q.

Example 7

medium
Translate to symbols: 'It is raining or it is snowing'. Use RR for raining, SS for snowing.

Example 8

easy
If both PP and QQ are true, what is P∨QP \vee Q (inclusive or)?

Example 9

medium
If P∨QP\vee Q is true and PP is false, what must QQ be?

Example 10

hard
Out of 80 students, 50 take Spanish, 35 take French, and 20 take both. How many take Spanish OR French?

Example 11

easy
If PP is false and QQ is true, what is P∨QP \vee Q?

Example 12

medium
Is P∨QP \vee Q logically equivalent to Q∨PQ \vee P?

Example 13

medium
If P∨QP\vee Q is false, what are PP and QQ?

Example 14

medium
Evaluate 'P∨QP \vee Q' where PP: '77 is even' and QQ: '77 is prime'.

Example 15

easy
In set notation, 'x∈Ax \in A OR x∈Bx \in B' is equivalent to x∈x \in ____ .

Example 16

easy
If PP is true and QQ is false, what is P∨QP \vee Q?

Example 17

medium
Solve x2−x−6=0x^2 - x - 6 = 0 and write the solution as a disjunction.

Example 18

medium
Find all real xx satisfying 'x<−1x < -1 or x>3x > 3', and express as a union of intervals.

Example 19

hard
Among the 16 possible truth functions of two variables, how many are true on at least one of the four rows? (Disjunction is one of them.)

Example 20

medium
Build the truth table column for ¬P∨Q\neg P \vee Q and identify the familiar connective it equals.
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