//M0//QN1//SUB//DL0

òu (x + 1, y – 2) = (3, 1) íkku, x yLku y þkuÄku.

//X

ynª, (x + 1, y – 2) = (3, 1) nkuðkÚke,
x + 1 = 3, y – 2 = 1 ÚkkÞ.
\ x = 2, y = 3

//M0//QN2//SUB//DL0

òu P = {a, b, c}, Q = {r}, íkku P × Q yLku Q × P þkuÄku. þwt yk fkíkuorÍÞ økwýkfkh Mk{kLk Au ?

//X

P × Q = {a, b, c} × {r}
= {(a, r), (b, r), (c, r)}...(1)
Q × P = {r} × {a, b, c}
= {(r, a), (r, b), (r, c)}...(2)
Ãkrhýk{ (1) yLku (2) ÃkhÚke òuE þfkÞ Au fu, ¢{Þwõík òuz (a, r) yLku (r, a) Mk{kLk Lk nkuðkÚke yk fkíkuorÍÞ økwýkfkh Mk{kLk LkÚke.

//M0//QN3//SUB//DL0

òu A = {1, 2, 3}, B = {3, 4}, C = {4, 5, 6}, íkku Lke[uLkk øký þkuÄku. (i) A × (B ∩ C) (ii) (A × B) ∩ (A × C) (iii) A × (B ∪ C) (iv) (A × B) ∪ (A × C)

//X

(i) A × (B ∩ C) {kxu,
B ∩ C = {4}
\ A × (B ∩ C) = {1, 2, 3} × {4}
= {(1, 4), (2, 4), (3, 4)}
(ii) (A × B) ∩ (A × C)
A × B = {1, 2, 3} × {3, 4}
= {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3),
(3, 4)}........(1)
A × C = {1, 2, 3} × {4, 5, 6}
= {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5),
(2, 6), (3, 4), (3, 5), (3, 6)}........(2)
\ (A × B) ∩ (A × C) = {(1, 4), (2, 4), (3, 4)}
(iii) A × (B ∪ C)
B ∪ C = {3, 4, 5, 6}
\ A × (B ∪ C) = {1, 2, 3} × {3, 4, 5, 6}
= {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3),
(2, 4), (2, 5), (2, 6), (3, 3),
(3, 4), (3, 5), (3, 6)}
(iv) (A × B) ∪ (A × C)
Ãkrhýk{ (1) yLku (2)Lkku WÃkÞkuøk fhíkkt,
(A × B) ∪ (A × C)
= {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3),
(2, 4), (2, 5), (2, 6), (3, 3),
(3, 4), (3, 5), (3, 6)}

//M0//QN4//SUB//DL0

òu P = {1, 2}, íkku P × P × P þkuÄku.

//X

P × P × P = {1, 2} × {1, 2} × {1, 2}
= {(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2),
(2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)}

//M0//QN5//SUB//DL0

òu R ðkMíkrðf MktÏÞkykuLkku øký nkuÞ, íkku R × R yLku R × R × R þwt Ëþkoðþu ?

//X

fkíkuorÍÞ økwýkfkh R × R = {(x, y) : x, y ∈ R} yu rîÃkrh{kýeÞ Þk{-Mk{ík÷Lkk «íÞuf ®çkËwLkwt rLkYÃký Ëþkoðu Au.
fkíkuorÍÞ økwýkfkh R × R × R = {(x, y, z) : x, y, z ∈ R} yu rºkÃkrh{kýeÞ Þk{-Mk{ík÷Lkk «íÞuf ®çkËwLkwt rLkYÃký Ëþkoðu Au.

//M0//QN6//SUB//DL0//EQ

òu A × B = {(p, q), (p, r), (m, q), (m, r)}, íkku A yLku B þkuÄku.

//X

øký A = A × BLke Ëhuf ¢{Þwõík òuzLkku «Úk{ ½xf
\ A = {p, m}
øký B = A × BLke Ëhuf ¢{Þwõík òuzLkku çkeòu ½xf
\ B = {q, r}

//M0//QN7//SUB//DL0//EQ

òu = , íkku x yLku y þkuÄku.

//X

ynª, =
\ + 1 = , y – =
\ = , =
\ x + 3 = 5 , 3y – 2 = 1
\ x = 2 , 3y = 3
\ y = 1

//M0//QN8//SUB//DL0

òu øký A{kt 3 ½xfku nkuÞ yLku B = {3, 4, 5}, íkku A × BLkk ½xfkuLke MktÏÞk þkuÄku.

//X

ynª, n(A) = 3, n(B) = 3
\ n(A × B) = n(A) ⋅ n(B) = 3 × 3 = 9 ÚkkÞ.

//M0//QN9//SUB//DL0

òu G = {7, 8}, H = {5, 4, 2}, íkku G × H yLku

H × G þkuÄku.

//X

G × H = {7, 8} × {5, 4, 2}
= {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}
H × G = {5, 4, 2} × {7, 8}
= {(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)}

//M0//QN10//SUB//DL0

Lke[u ykÃku÷kt rðÄkLkku{ktÚke fÞwt rðÄkLk MkíÞ Au yLku fÞwt rðÄkLk yMkíÞ Au, íku sýkðku íkÚkk yMkíÞ rðÄkLk MkíÞ çkLku íku VheÚke ÷¾ku.(i) òu P = {m, n}, Q = {n, m}, íkku P × Q = {(m, n), (n, m)}.

//X

ykÃku÷ rðÄkLk yMkíÞ Au.
MkíÞ rðÄkLk : P × Q = {(m, n), (m, m), (n, n), (n, m)}
(ii) òu A yLku B yrhõík øký nkuÞ, íkku ßÞkt, x ∈ A íkÚkk y ∈ B nkuÞ íkuðe ík{k{ ¢{Þwõík òuz (x, y)Úke çkLkíkku yrhõík øký A × B Au.
ykÃku÷ rðÄkLk MkíÞ Au.
(iii) òu A = {1, 2}, B = {3, 4}, íkku A × (B ∩ f) = f
ykÃku÷ rðÄkLk MkíÞ Au.
ynª, B ∩ f = {3, 4} ∩ f = f Au.
\ A × (B ∩ f) = {1, 2} × f = f ÚkkÞ.
fu{ fu, çktLku{ktÚke fkuE Ãký yuf øký òu ¾k÷e øký nkuÞ, íkku íku{Lkku fkíkuorÍÞ økwýkfkh ¾k÷e øký s ÚkkÞ.

//M0//QN11//SUB//DL0

òu A = {–1, 1}, íkku A × A × A {u¤ðku.

//X

A × A × A = {–1, 1} × {–1, 1} × {–1, 1}
= {(–1, –1, –1), (–1, –1, 1), (–1, 1, –1), (–1, 1, 1), (1, –1, –1), (1, –1, 1),
(1, 1, 1), (1, 1, –1)}

//M0//QN12//SUB//DL0

òu A × B = {(a, x), (a, y), (b, x), (b, y)}, íkku A yLku B þkuÄku.

//X

ykÚke, (a, x) ∈ A × B
\ a ∈ A, x ∈ B
(a, y) ∈ A × B
\ a ∈ A, y ∈ B
(b, x) ∈ A × B
\ b ∈ A, x ∈ B
ykÚke, øký A = A × BLke «íÞuf ¢{Þwõík òuzLkku «Úk{ ½xf
= {a, b}
øký B = A × BLke «íÞuf ¢{Þwõík òuzLkku çkeòu ½xf
= {x, y}

//M0//QN13//SUB//DL0

Äkhku fu, A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6},

D = {5, 6, 7, 8}, íkku Lke[uLkkt Ãkrhýk{ku [fkMkku :(i) A × (B ∩ C) = (A × B) ∩ (A × C)

//X

zk.çkk. A × (B ∩ C) {kxu,
B ∩ C = f
\ A × (B ∩ C) = {1, 2} × f = f...(1)
s.çkk. (A × B) ∩ (A × C)

A × B = {1, 2} × {1, 2, 3, 4}

= {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1),

(2, 2), (2, 3), (2, 4)}

A × C = {1, 2} × {5, 6}

= {(1, 5), (1, 6), (2, 5), (2, 6)}

\ (A × B) ∩ (A × C) = f...(2)
\ Ãkrhýk{ (1) yLku (2) ÃkhÚke
zk.çkk. = s.çkk.
(ii) A × C yu B × DLkku WÃkøký Au.
ynª, A × C = {(1, 5), (1, 6), (2, 5), (2, 6)}

B × D = {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6),
(3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8)}

ynª, òuE þfkÞ Au fu, A × CLke Ëhuf ¢{Þwõík òuz B × D{kt ykðu÷ nkuðkÚke A × C yu B × DLkku WÃkøký Au.

//M0//QN14//SUB//DL0

òu A = {1, 2}, B = {3, 4}, íkku A × B þkuÄku. A × BLku fux÷k WÃkøkýku nþu ? íku ík{k{ WÃkøkýkuLke ÞkËe çkLkkðku.

//X

A × B = {1, 2} × {3, 4} = {(1, 3), (1, 4), (2, 3), (2, 4)}
ynª, n(A × B) = 4 nkuðkÚke íkuLkk WÃkøkýkuLke MktÏÞk 24 = 16 Úkþu.
  • WÃkøkýku : f, A × B, {(1, 3)}, {(1, 4)}, {(2, 3)},
    {(2, 4)}, {(1, 3), (1, 4)}, {(1, 3), (2, 3)}, {(1, 3), (2, 4)}, {(1, 4), (2, 3)}, {(1, 4),
    (2, 4)}, {(2, 3), (2, 4)}, {(1, 3), (1, 4),
    (2, 3)}, {(1, 3), (1, 4), (2, 4)}, {(1, 4),
    (2, 3), (2, 4)}, {(2, 3), (2, 4), (1, 3)}

//M0//QN15//SUB//DL0

òu n(A) = 3, n(B) = 2 nkuÞ íkuðk çku øký A yLku B nkuÞ yLku r¼LLk ½xfku x, y, z {kxu (x, 1), (y, 2),

(z, 1) yu A × BLkk ½xfku nkuÞ, íkku A yLku B þkuÄku.

//X

ynª, (x, 1), (y, 2), (z, 1) ∈ A × B nkuðkÚke,
A = {x, y, z}, B = {1, 2} ÚkkÞ, fkhý fu n(A) = 3, n(B) = 2 ykÃku÷ Au.

//M0//QN16//SUB//DL0

òu fkíkuorÍÞ økwýkfkh A × ALkk ½xfkuLke MktÏÞk 9 nkuÞ yLku íku{ktLkk çku ½xfku (–1, 0) yLku (0, 1) nkuÞ, íkku A þkuÄku íkÚkk A × ALkk çkkfeLkk ½xfku ÷¾ku.

//X

ynª, n(A × A) = 9 \ n(A) = 3 ÚkkÞ íkÚkk (–1, 0), (0, 1) ∈ A × A nkuðkÚke A = {–1, 0, 1} ÚkkÞ.
{kxu, A × ALkk çkkfeLkk ½xfku (–1, 1), (–1, –1),
(0, –1), (0, 0), (1, –1), (1, 0) yLku (1, 1) Au.