Venn diagrams and valid conclusions · Logic

Many logic questions give statements about groups of people or things and ask you to draw a Venn diagram. Each kind of statement has its own picture:

“All $P$ are $Q$ ”: the circle $P$ sits inside $Q$ . The same picture fits “every $P$ is a $Q$ ”, “only $Q$ s are $P$ s”, “there is no $P$ who is not a $Q$ ” and “if $x$ is a $P$ , then $x$ is a $Q$ ”.
“No $P$ is $Q$ ”: the circles are apart. Nothing is in both.
“Some $P$ are $Q$ ”: the circles overlap, and at least one member is in the overlap.

All P are QP inside Q (also: only Qs are Ps)

No P is QThe circles don't meet

Some P are QThe overlap has at least one member

Is the conclusion valid?

A conclusion such as “if $x$ is a $Q$ , then $x$ is a $P$ ” is valid when it’s true for everyone the diagram allows. To test it, look for a region where the “if” part is true but the “then” part is false. If there is such a region, someone could be standing there, and the conclusion is not valid. If there’s no such region, it is valid.

Try it

Valid or not valid?Tap a region to put X there

·X is a captain ⇒ X is a footballer
·X is a footballer ⇒ X is a captain
·X is not a footballer ⇒ X is not a captain
·X is not a captain ⇒ X is not a footballer

All captains are footballersthe statement

“All captains are footballers.” Every captain is inside the footballers circle; some footballers are not captains. Tap a region to put X there.

Put X in each region in turn and watch which conclusions break. Then switch on “Test every region”. In the “All” diagram, the two valid conclusions are the statement itself and its contrapositive, just as in the last lesson.

Two statements together

When two statements share a set, draw them on one diagram. “All $A$ are $B$ ” and “all $B$ are $C$ ” give three circles, each inside the next. Then “all $A$ are $C$ ” is valid: $A$ is inside $B$ , which is inside $C$ .

All A are B; all B are CA inside B inside C, so all A are C

Worked example · WAEC 2022

WAEC 2022 · Paper 2 · Q7 (b)

Consider the following statements:

$P$ : There is no Mathematics student who is not clever. $Q$ : Every Physics student studies Mathematics.

(i) Draw a Venn diagram to represent the statements. (ii) Deduce whether the following statements are valid or not valid. (I) Every clever student is a Physics student. (II) All Physics students are clever. (III) Every Mathematics student studies Physics.

Turn P into a picture
No Mathematics student is outside the clever students, so every Mathematics student is clever: $M$ (Mathematics) sits inside $C$ (clever).
Think first. “No Mathematics student who is not clever”: which set goes inside which?
Turn Q into a picture
Every Physics student studies Mathematics: $Ph$ (Physics) sits inside $M$ .
Think first. Where does the Physics circle go?
(i) Draw one diagram
Three circles, each inside the next, in a universal set of students:
Physics ⊂ Maths ⊂ Clever
Think first. How many circles, and how are they arranged?
(ii)(I) Every clever student is a Physics student
Yes: a clever student outside the Maths circle, for example. Not valid.
Think first. Is there a region inside Clever but outside Physics?
(II) All Physics students are clever
No: Physics is inside Maths, which is inside Clever. Valid.
Think first. Can anyone be inside Physics but outside Clever?
(III) Every Mathematics student studies Physics
Yes: the ring between the Physics and Maths circles. Not valid.
Think first. Is there a region inside Maths but outside Physics?

Your turn

WAEC 2020 · Paper 2 · Q1 (a)

(a)
Consider the statements:

$p$ : Some teachers in a school are graduates. $q$ : Only graduate teachers received National honours.

(i) Represent this information in a Venn diagram. (ii) If Mr. Sowah is a teacher in the school, determine whether or not the following conclusions are valid or not valid. (I) Mr. Sowah is a graduate ⇒ Mr. Sowah received National honours. (II) Mr. Sowah received National honours ⇒ Mr. Sowah is a graduate. (III) Mr. Sowah did not receive a National honour ⇒ Mr. Sowah is not a graduate.
Model answer
$U$ = teachers in the school, $G$ = graduate teachers, $N$ = teachers who received National honours. "Some teachers are graduates" puts $G$ inside $U$ but not filling it; "only graduates received honours" puts $N$ inside $G$ . So a teacher in $N$ must be in $G$ (II is valid), but a teacher in $G$ need not be in $N$ (I and III are not valid).

Worked solution (try it first)

(a)(i)

Let the universal set be the teachers in the school, $G$ the graduate teachers and $H$ the teachers who received National honours. "Some teachers are graduates": draw $G$ inside the rectangle, leaving room outside it. "Only graduate teachers received National honours": draw $H$ inside $G$ .

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