Solution to Solution: a) p∧q b)p∧¬q c)¬p∧¬ q d)p∨q e)p→ q f)(p∨q)∧(p→ ¬ q) g)p↔q - Sikademy
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Archangel Macsika

Solution: a) p∧q b)p∧¬q c)¬p∧¬ q d)p∨q e)p→ q f)(p∨q)∧(p→ ¬ q) g)p↔q

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Let N = 250 be the total members of the society voted for X, Y, Z. Let n(X), n(Y), n(Z) denote the number of candidates voted for X, Y, Z respectively. Then,

\begin{aligned} N &= n(X \cup Y \cup Z) + n(X \cup Y \cup Z)'\qquad (\text{Voted + Not voted})\\ &= n(X) + n(Y) + n(Z) - n(X \cap Y)-n(X \cap Z)- n(Y \cap Z) \\ &\qquad\qquad\qquad\qquad\qquad+ n(X \cap Y \cap Z)+n(X \cup Y \cup Z)'\\ 250 &= n(X) + n(Y) + n(Z) - n(X \cap Y)-n(X \cap Z) - n(Y \cap Z)\\ &\qquad\qquad\qquad\qquad\qquad + n(X \cup Y \cup Z)' \qquad\qquad\qquad\qquad\qquad(1)\\ &(\text{Since each member may vote for either one or two candidates}\\ &~~~~~~~~~~~~n(X \cap Y \cap Z)=0)\\ \end{aligned}

12~ \text{voted for X and Y, ~} 14~ \text{voted for X and Z, i.e.,}\\ \begin{aligned} n(X \cap Y) &=12 \qquad\qquad\quad (2)\\ n(X \cap Z) &=14\qquad\qquad\quad (3)\\\\ \end{aligned}\\ 59~ \text{voted for Y only and~ } 37~ \text{voted for Z only, i.e.,}\\ \begin{aligned} n(Y) - n(X \cap Y) - n(Y \cap Z) &=59 \qquad\qquad\quad (4)\\ n(Z) - n(X \cap Z) - n(Y \cap Z) &=37\qquad\qquad\quad (5)\\\\ \end{aligned}\\ 147~ \text{voted for X or Y or both but not for Z }\\ 102~ \text{voted for Y or Z or both but not for X, i.e.,}\\ \begin{aligned} n(X) + n(Y) - n(X \cap Y) - n(X \cap Z) - n(Y \cap Z) &=147 \qquad\qquad (6)\\ n(Y) + n(Z) - n(Y \cap Z) - n(X \cap Z) - n(X \cap Y) &=102\qquad\qquad(7)\\\\ \end{aligned}\\

Adding (4) and (5), we get

n(Y) + n(Z) - n(X \cap Y) - n(X \cap Z) - 2\cdot n(Y \cap Z) = 96 \qquad\quad (8)


Subtracting (8) from (7), we get n(Y \cap Z) = 6.


From (4), (5), (6)

\begin{aligned} n(Y) &= 59 + n(X\cap Y) + n(Y \cap Z) = 59+12+6=77\\ n(Z) &= 37 + n(X\cap Z) + n(Y \cap Z) = 37+14+6=57\\ n(X) & = 147- n(Y) + n(X \cap Y)+ n(X \cap Z) + n(Y \cap Z)\\ &=147 -77+12+14+6 = 102 \end{aligned}

i)



ii) Number of candidates who did not vote = n(X \cup Y \cup Z)'

From (1),

\begin{aligned} n(X \cup Y \cup Z)' &= 250 - (n(X) + n(Y) + n(Z) - n(X \cap Y)-n(X \cap Z) - n(Y \cap Z))\\ &= 250-(102+77+57-12-14-6)\\ & = 46 \end{aligned}

iii) Number of members voted for X only = n(X) - n(X \cap Y) - n(X \cap Z) = 102-12-14 = 76


iv) Number of votes for X only = 76,

Number of votes for Y only = 59,

Number of votes for Z only = 37.

Therefore, X won the election.


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