Superheroic Symbolizations Answers

Answers to Superheroic Symbolizations

Sx: is a superhero
Fx: can fly
Tx: wears tights
Mx: wears a mask
b: Batman
c: Catwoman

Following the conventions adopted by the Logic Daemon, I use @ for the universal quantifier and $ for the existential quantifier. This ensures that the quantifiers will appear on your screen and when you print these out.

1. There are no superheroes.
    @x~Sx
    ~$xSx
2. There are some superheroes.
    $xSx
3. All superheroes wear tights.
    @x(Sx -> Tx)
4. Not all superheroes wear tights.
    ~@x(Sx -> Tx)
    $x(Sx & ~Tx)
5. Some superheroes wear tights.
    $x(Sx & Tx)
6. Some superheroes do not wear tights.
    $x(Sx & ~Tx)
7. Only superheroes wear tights.
    @x(Tx -> Sx)
8. No superheroes wear tights.
    @x(Sx -> ~Tx)
    ~$x(Sx & Tx)
9. Superheroes wear tights.
    @x(Sx -> Tx)
10. All masked superheroes wear tights.
    @x(Sx & Mx -> Tx)
11. Not all masked superheroes wear tights.
    ~@x(Sx & Mx -> Tx)
12. Some masked superheroes wear tights.
    $x((Sx & Mx) & Tx)    {alternate parentheses OK}
13. Some masked superheroes do not wear tights.
    $x((Sx & Mx) & ~Tx)    {alternate parentheses OK}
14. Not every superhero can fly, though they all wear tights.
    ~@x(Sx -> Fx) & @x(Sx -> Tx)
    $x(Sx & ~Fx) & @x(Sx -> Tx)
15. Batman, who is a superhero, cannot fly.
    Sb & ~Fb
16. Catwoman is not a superhero but she wears a mask.
    ~Sc & Mc
17. Neither Batman nor Catwoman can fly, but they both wear masks.
    ~(Fb v Fc) & (Mb & Mc)
    (~Fb & ~Fc) & (Mb & Mc)
18. Some superheroes wear tights but not masks.
    $x((Sx & Tx) & ~Mx)    {alternate parentheses OK}
19. Some superheroes who wear tights also wear masks.
    $x((Sx & Tx) & Mx)    {alternate parentheses OK}
20. Superheroes who wear tights also wear masks.
    @x(Sx & Tx -> Mx)
21. Superheroes, who wear tights, also wear masks.
    @x(Sx -> Tx & Mx)
22. Unless a superhero wears a mask he does not wear tights.
    @x(Sx -> Mx v ~Tx)
    @x(Sx -> (~Mx -> ~Tx))
23. No superhero who wears a mask fails to wear tights.
    @x(Sx & Mx -> Tx)
    ~$x((Sx & Mx) & ~Tx)
24. All superheroes either wear masks or wear tights.
    @x(Sx -> Mx v Tx)
25. Batman is a superhero who wears tights.
    Sb & Tb

26. Batman is not identical to Catwoman

~b=c {use of the non-identity sign is also OK}