A linear programming problem (LPP) is a problem of maximising or minimising a linear function of variables subject to certain linear conditions. The objective function is the linear function Z = ax + by that is to be optimised (maximised or minimised). The constraints are the linear inequalities (or equations) in the variables that restrict the values the variables can take; together with the non-negativity restrictions (x >= 0, y >= 0) they define the region of possible solutions.

Z = ax + by (objective function)

The feasible region is the common region determined by all the constraints (including the non-negativity restrictions) of a linear programming problem; it is the set of all points that satisfy every constraint. A feasible solution is any point that lies in the feasible region. An optimal solution is a feasible solution that gives the maximum or minimum value of the objective function, whichever is required.

A feasible region is said to be bounded if it can be enclosed within a circle (or a rectangle) of finite radius, that is, it does not extend to infinity. It is unbounded if it extends indefinitely (to infinity) in some direction and cannot be enclosed. For a bounded feasible region, the objective function always has both a maximum and a minimum value at the corner points. For an unbounded region, a maximum or a minimum may not exist and must be checked separately.

Conditional probability is the probability of occurrence of an event A given that another event B has already occurred; it is written as P(A | B). Its formula is P(A | B) = P(A intersection B)/P(B), provided P(B) is not zero. Here P(A intersection B) is the probability that both A and B occur, and P(B) is the probability of B.

P(A | B) = P(A intersection B)/P(B)

Two events A and B are said to be independent if the occurrence of one does not affect the probability of the occurrence of the other, that is, P(A | B) = P(A) and P(B | A) = P(B). For independent events, the multiplication rule takes the simple form P(A intersection B) = P(A) x P(B). This is used as the test for independence: if P(A intersection B) = P(A) P(B), the events are independent.

P(A intersection B) = P(A) x P(B)

A random variable is a real-valued function whose value is determined by the outcome of a random experiment; it assigns a number to each outcome of the sample space (for example, the number of heads when two coins are tossed). A probability distribution of a random variable is a table (or description) that lists all the possible values of the random variable together with their corresponding probabilities; the sum of all these probabilities is always 1.

sum of all p(x) = 1

The two reproductive whorls of a flower are the androecium and the gynoecium. The androecium is the male reproductive whorl made up of stamens; each stamen has an anther that produces pollen grains (which contain the male gametes). The gynoecium (pistil) is the female reproductive whorl made up of one or more carpels; each carpel has an ovary containing ovules (which contain the female gamete, the egg), a style and a stigma that receives the pollen.

A typical mature embryo sac (female gametophyte) is 7-celled and 8-nucleate. At the micropylar end there is the egg apparatus consisting of one egg cell and two synergids (with special thickenings called filiform apparatus). At the chalazal end there are three antipodal cells. In the centre there is a large central cell containing two polar nuclei. Thus, although there are 8 nuclei, there are only 7 cells because the central cell has two nuclei.

Pollination is the transfer of pollen grains from the anther to the stigma of a flower. Self-pollination (autogamy) is the transfer of pollen from the anther to the stigma of the same flower or another flower of the same plant; it maintains parental characters. Cross-pollination (allogamy) is the transfer of pollen to the stigma of a flower on a different plant of the same species; it brings about variation. Cross-pollination is carried out by agents such as wind, water, insects and birds.

After fertilisation: (1) the zygote develops into the embryo; (2) the primary endosperm nucleus develops into the endosperm, which nourishes the developing embryo; (3) the ovule develops into the seed and its integuments harden into the seed coat; and (4) the ovary develops into the fruit and its wall becomes the pericarp. Other parts of the flower such as the petals, stamens, style and stigma usually wither and fall off.

Apomixis is the formation of seeds without fertilisation; it is a form of asexual reproduction that mimics sexual reproduction (for example, in some species of grasses and Citrus). The seeds so formed are genetically identical to the parent. Polyembryony is the occurrence of more than one embryo in a single seed; it is commonly seen in Citrus and mango, where the extra embryos may develop from the cells of the nucellus.

Many flowering plants have devices to encourage cross-pollination (outbreeding) because continued self-pollination leads to inbreeding depression, and cross-pollination introduces genetic variation, which is important for adaptation and evolution. Two such outbreeding devices are: (1) unisexuality - the male and female flowers are separate (or on separate plants), so self-pollination is not possible; and (2) different timing of maturation of the anther and the stigma (dichogamy - protandry or protogyny), so that they are not receptive at the same time. Another device is self-incompatibility.

The human male reproductive system consists of: a pair of testes (the primary sex organs, which produce sperms and the male hormone testosterone), housed in the scrotum; a system of ducts, namely the epididymis, vas deferens, and urethra, that carry and store the sperms; and the accessory glands - the seminal vesicles, the prostate gland and the bulbourethral (Cowper's) glands - whose secretions form the seminal fluid. The penis is the copulatory organ.

The human female reproductive system consists of: a pair of ovaries (the primary sex organs, which produce ova/eggs and the hormones oestrogen and progesterone); a pair of oviducts (fallopian tubes), which carry the egg and are the usual site of fertilisation; the uterus (womb), where the embryo develops; the cervix, which connects the uterus to the vagina; and the vagina, which is the copulatory canal and birth canal.

Oogenesis is the process of formation of the mature female gamete (ovum) from the germ cells in the ovary; it begins during the embryonic stage. An oogonium forms a primary oocyte, which undergoes the first meiotic division to form a secondary oocyte and a small first polar body; the second meiotic division (completed only after fertilisation) gives one large ovum and a second polar body. Thus, one primary oocyte forms only one functional ovum, whereas in spermatogenesis one primary spermatocyte forms four functional sperms.

A human sperm is a microscopic, motile cell with three main parts: (1) a head, which contains the haploid nucleus and is capped by the acrosome (containing enzymes that help the sperm penetrate the egg); (2) a middle piece, which contains many mitochondria that provide the energy for movement; and (3) a tail (flagellum), which lashes to propel the sperm. A plasma membrane covers the whole body of the sperm.

Fertilization is the fusion of a sperm with the ovum to form a diploid zygote; it normally takes place in the ampullary region of the fallopian tube. The zygote then undergoes repeated cell divisions (cleavage) as it moves down the tube to form a hollow ball of cells called the blastocyst. Implantation is the attachment and embedding of the blastocyst into the thick, prepared wall (endometrium) of the uterus, where the embryo then develops further.

The placenta is a special disc-shaped structure that connects the developing foetus (embryo) to the wall of the mother's uterus; it is formed from the tissues of both the foetus and the mother and is connected to the foetus by the umbilical cord. Functions: (1) it allows the exchange of materials - it supplies oxygen and nutrients from the mother to the foetus and removes carbon dioxide and wastes from the foetus; and (2) it acts as an endocrine gland, secreting hormones such as hCG, oestrogen and progesterone to maintain the pregnancy.

Oestrogen is secreted mainly by the growing ovarian follicle; it stimulates the repair and thickening of the uterine lining after menstruation, develops secondary sexual characters, and (at high levels) triggers the LH surge that causes ovulation. Progesterone is secreted mainly by the corpus luteum after ovulation; it maintains and further thickens the uterine lining (endometrium) to prepare it for implantation and, along with other hormones, maintains pregnancy if fertilisation occurs.

Reproductive health means a total well-being in all aspects of reproduction - physical, emotional, behavioural and social. A reproductively healthy society is one whose people have physically and functionally normal reproductive organs and healthy attitudes towards reproduction. Two measures to achieve it are: (1) creating awareness through education about reproduction, safe practices and available facilities (for example the RCH - Reproductive and Child Health - programmes); and (2) providing medical facilities and care for pregnancy, contraception, birth control and treatment of sexually transmitted infections.