Syllogisms

Name:

Class:

Date:

Show your work for each problem.

1.
All rectangles are parallelograms. All squares are rectangles. Therefore all squares are parallelograms. What is the middle term?
1. a)Quadrilaterals
2. b)Rectangles
3. c)Parallelograms
4. d)Squares
2.
A Venn diagram has three overlapping circles $S$ , $M$ and $P$ . How many regions does it have, counting the region outside all three circles?
3.
Remember: A = All $S$ are $P$ , E = No $S$ are $P$ , I = Some $S$ are $P$ , O = Some $S$ are not $P$ . Which list gives the codes of these four statements in order? (1) Some rectangles are not squares. (2) All squares are rectangles. (3) Some rectangles are squares. (4) No triangle is a square.
1. a)O, E, I, A
2. b)I, A, O, E
3. c)O, A, I, E
4. d)E, A, I, O
4.
All $14$ members of the chess club are also in the math club. The math club has $25$ members. How many math club members are not in the chess club?
5.
No prime number greater than $2$ is even. All numbers in the set $T$ are prime numbers greater than $2$ . Which conclusion follows?
1. a)All even numbers are in $T$
2. b)Nothing follows
3. c)Some numbers in $T$ are even
4. d)No number in $T$ is even
6.
All multiples of $6$ are multiples of $3$ . All multiples of $12$ are multiples of $6$ . Therefore all multiples of $12$ are multiples of $3$ . Name the middle term, then follow the number $60$ (a multiple of $12$ ) through the chain. What is $60 \div 3$ ?
7.
All poets are dreamers. Some dreamers are engineers. Therefore some poets are engineers. How do you judge this argument?
1. a)Valid, but one premise is false
2. b)Valid and sound
3. c)Invalid, because a syllogism may not use the word some
4. d)Invalid: the poets and the engineers may be different dreamers
8.
A class has $60$ students. No member of the swim team is on the soccer team. The swim team has $16$ members and the soccer team has $22$ . How many students are on neither team?
9.
A class has $32$ students. All $8$ members of the chess club are in the math club, which has $15$ members. No member of the math club is in the drama club, which has $10$ members. How many students are in none of the three clubs?
10.
All multiples of $4$ are even. All multiples of $6$ are even. Therefore all multiples of $6$ are multiples of $4$ . Show that the argument is invalid with a counterexample. What is the smallest positive multiple of $6$ that is not a multiple of $4$ ?
11.
All multiples of $4$ are multiples of $8$ . $12$ is a multiple of $4$ . Therefore $12$ is a multiple of $8$ . Which description is correct?
1. a)Valid but not sound: the first premise is false
2. b)Invalid, because the conclusion is false
3. c)Invalid: undistributed middle
4. d)Valid and sound
12.
An orchestra has $24$ members. All $9$ students who play the violin are in the orchestra. $4$ of the violinists sing in the choir, and $7$ other orchestra members also sing in the choir. How many orchestra members do not sing in the choir?
13.
Test the syllogism No $M$ are $P$ , Some $S$ are $M$ , so Some $S$ are not $P$ with the 8 region diagram (region 4 = inside $S$ and $M$ only, region 6 = inside $M$ and $P$ only, region 7 = inside all three). What is the code of the conclusion?
14.
In the three circle Venn diagram for $S$ , $M$ and $P$ , you shade the premises All $S$ are $M$ and No $M$ are $P$ . How many of the 8 regions are shaded in total?
15.
The statements $p \to q$ and $q \to r$ are true, and $r$ is false. Which statement must be true?
1. a) $q$
2. b) $p$
3. c) $\neg p$
4. d) $r \to p$
16.
Premises: If $n$ is divisible by $20$ , then $n$ is divisible by $10$ . If $n$ is divisible by $10$ , then the last digit of $n$ is $0$ . If the last digit of $n$ is $0$ , then $n$ is divisible by $5$ . The chain rule gives: if $n$ is divisible by $20$ , then $n$ is divisible by $5$ . Check it with $n = 140$ , then show that the converse does not follow. What is the smallest positive number divisible by $5$ but not by $20$ ?
17.
A sports club has $70$ members and $45$ of them swim. All $20$ tennis players are swimmers. $12$ tennis players also run, $8$ swimmers who do not play tennis run, and $15$ members who do not swim run. How many members neither swim nor run?
18.
Show that Some $S$ are $M$ , Some $M$ are $P$ , so Some $S$ are $P$ is invalid. Use $S = \{1, 2\}$ , $M = \{2, 3\}$ and $P = \{3, 4\}$ . How many elements do $S$ and $P$ have in common?