Page 24 - Algebra 1

P. 24

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Explanation
1. P = {π΄ππ π‘ππ₯π‘πππππ ππ€πππ ππ¦ πππ}
R = {(π, π): π πππ π hππ£π π‘hπ π πππ ππ’ππππ ππ πππππ }
R is reflexive on A since (p, p) β R
R is symmetric on A since books p and q have the same number of page (p, q) β R.
Also, q and p have the same number of pages. So, (q, p) β R
Therefore, R is symmetric on A
R is transitive on A, since books p, q, and r have the same number of pages (p, q) β R and (q, r) βR.So,(p,r) βR
R is reflexive, symmetric, and transitive on A. So, itβs equivalent on A.
2. P={(π,π):πβ€π4},where(p,q)βR
Reflexive:
1 β€ (1)4 doesnβt hold true.
So, R is not reflexive.
Symmetry:
β1 β€ (3)4 hold true and 3 β€ (β 1)4 doesnβt holds true. (3, β1) Ι R and (β1, 3) β R
So, R is not symmetric.
Transitive:
4 β€ (β 5)4 hold true and β5 β€ (1)4 holds true. 4 β€ (1)4 doesnβt hold true.
So, R is not transitive.
Therefore, R is neither symmetric, nor reflexive nor transitive.
3. Relation from Q to P
P = {1,2} and Q = {π,π,π}
R = {(π, 1), (π, 2), (π, 1), (π, 2), (π, 1), (π, 2)}
4. R is not reflexive since (3, 3), (4, 4) Ι R
R is symmetric since (3, 4) β R and (4, 3) β R
R is not transitive since (3, 4) β R but (4, 5), (3, 4) Ι R, so R is transitive R is not equivalence since itβs only symmetric
5. Number of relations possible from P to Q = 2rs Number of relations possible = 2(6 Γ 3) = 218
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Page 23 of 54
ALGEBRA
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